Words can change their meaning. So whether Leupp did say/write these words "final solution" would be the first thing to ascertain. Then we should look for what he meant by that. After Hitler, "final solution" is usually quite specific in meaning industrialised mass murder and cold blooded genocide.

Some argue that the treatment of native Americans would constitute a kind of genocide. Some argue that this would be on a level comparable to what happened to the Jews of Europe. The first is much easier to argue for, the second position usually not, by far not. It is often done out of despicable motives to relativise the Holocaust. Even if that is not the goal, instead trying to paint ("white") American history as terrible as possible, it still is Holocaust relativism.

What is this here around such a quote:

In the three or four decades after treaty-making was discontinued, the BIA demonised the tribal governments it had previously relied on to deliver treaties and focused on the improvable individual, whose individuality corresponded to a particular fragment of the tribal estate. The outcome was a two-way loss whereby culture and biology supplemented each other. As Senator Higgins put it in Congress: ‘It seems to me one of the ways of getting rid of the Indian question is just this of intermarriage, and the gradual fading out of the Indian blood; the whole quality and character of the aborigine disappears, they lose all of the traditions of the race’. Culturally, through what Lewis Meriam, author of the scathing report that heralded the end of the allotment programme, sarcastically dubbed ‘the magic in individual ownership of property’, Indians would be co-opted out of the tribe, which would be depleted accordingly, and into White society.

Lewis Meriam et al., The Problem of Indian Administration. Summary of Findings and Recommendations [aka the Meriam Report] (Washington, DC: Institute for Government Research, 1928), p. 7.

‘If we can watch our body of dependent Indians shrink even by one member at a time, we may congratulate ourselves that the complete solution is only a question of patience’: Francis E. Leupp [former Commissioner for Indian Affairs], The Indian and His Problem (New York: Charles Scribner’s Sons, 1910), p. 49.

_{–– Patrick Wolfe: "After the Frontier: Separation and Absorption in US Indian Policy", Settler Colonial Studies, 1:1, 13-51, 2011 DOI: 10.1080/2201473X.2011.10648800}

An earlier version of Leupp's text does use the word pair "final solution":

United States. Office of Indian Affairs / Annual report of the commissioner of Indian affairs, for the year 1905

but in the exact same context.

It seems the author is quoting this version out of preference. (But it looks as if Leupp just liked the word final a bit too much on stylistic grounds and revised the text later?) Quoting the earlier version surely changes the effect on the modern reader, but it seems doubtful that this accurately reflects the intended meaning at the time.

Leupp's "final/complete solution" was primarily Americanising Indians, making them 'good' citizens. Hitler's "final solution" was primarily killing.

One full text on this by Leupp is on archive org, to give this more context:

Incidentally to this programme, let us seek to make of the Indian an independent laborer as distinguished from one for whom the Government is continually straining to invent occupations. He can penetrate a humbug, even a benevolent humbug, as promptly as the next man; and when he sees the Government creating purely fictitious needs in order to find a pretext for giving him something to do, he despises the whole thing as a fraud, like the poor white whom some misguided philanthropist hires to carry a pile of bricks from one side of the road to the other and then back again. The employment bureau which I organized experimentally in the Southwest in 1905, and afterward extended to cover the whole Western country, is designed to gather up all the able-bodied Indians who, through the pinch of hunger it may be, have been moved to think that they would like to earn some money, and plant them upon private ranches, upon railroads, in mines, or wherever else in the outer world a dollar can be fairly earned by days' labor. The Supervisor in charge of the bureau at first scrutinizes their contracts with their employers, sees that their wages are paid them when due, and looks out for them if they fall ill ; but as soon as any of them show that they are able to attend to these things for themselves, he takes his hands off, and all who have been thus set upon their feet are given to understand that for whatever comes to them thereafter they will have themselves to thank.

Some one has styled this a policy of shrinkage, because every Indian whose name is stricken from a tribal roll reduces, by virtue of such emancipation, the dimensions of our red-race problem by a fraction— small, perhaps, but by no means negligible.

If we can watch our body of dependent Indians shrink even by one member at a time, we may congratulate ourselves that the complete solution is only a question of patience.

The process of general readjustment, though necessarily gradual, should be carried forward as fast as it can be with presumptive security for the Indian’s little possessions, and I should not let its educative value be obscured for a moment. The leading-strings which have tied the Indian to the Treasury ever since he began to own anything worth owning have been a curse to him. They have kept him an economic nursling long past the day when he ought to have been able to take a few steps without assistance, and left him an easy victim to such waves of civic heresy as sweep over the sparsely settled West from time to time.

_{–– Francis Ellington Leupp: "The Indian and his problem", 1910}

This "complete solution" bears in my eyes not the slightest resemblance to anything what is commonly understood as the "final solution" today.

As should be clear from the context within the one sentence quote already:

Earlier version:

If we can gradually watch our body of dependent Indians shrink, even by one member at a time, we may congratulate ourselves that the final solution is indeed only a question of a few years.

Later version:

If we can watch our body of dependent Indians shrink even by one member at a time, we may congratulate ourselves that the complete solution is only a question of patience.

The problem are the dependent Indians. 'Make them independent, problem gone.'

It is still not good, to the contrary, extremely patriarchical and rife with colonialist racist attitude, but in comparison almost benign.

Comparing even that 1905 text to German nazi racist doctrine seems a far stretch. The 1905 report starts with:

The commonest mistake made by his white wellwishers in dealing with the Indian is the assumption that he is simply a white man with a red skin. The next commonest is the assumption that because he is a non-Caucasian he is to be classed indiscriminately with other non-Caucasians, like the negro, for instance. The truth is that the Indian has as distinct an individuality as any type of man who ever lived, and he will never be judged aright till we learn to measure him by his own standards, as we whites would wish to be measured if some more powerful race were to usurp dominion over us.

Quite racist indeed, but not very eliminatory?

You might compare Leupp's writings with German antisemitism. Not that of 1941, but from the exact same time, at the turn of the century:

Theodor Fritsch, a central figure of the "Berlin Movement", called in his 1887 book "Anti-Semite Catechism" for anti-Semitism to be carried into all parties and organizations in order to gain a majority in the Reichstag and force the exclusion of Jews with a law. In the 28th edition of his "Handbook of the Jewish Question" of 1910 (originally: Theodor Fritsch: "Handbuch der Judenfrage", Hanseatische Druck- und Verlagsanstalt, Hamburg, 1907.) it was stated: "The last solution of the Jewish problem can only be found in a complete elimination of all Jews from Aryan ethnic life. A compromise is unacceptable".

The German Social Reform Party emerged in 1894 from the merger of two older anti-Semite parties of the Empire. Their program of 1899 demanded for the first time to solve the Jewish question "by complete segregation and (if the self-defence demands it) finally annihilation of the Jewish people". In 1914, the German People's Party also proclaimed in its founding program that the final "solution of the Jewish question" would become the "world question of the 20th century".

It might be easy to find an individual within the US administrative hierarchy making remarks "with such a sentiment". As the overarching question is

Q. To what degree was such a sentiment of a 'final solution' held by senior officials of the then US administration, the president of which was Theodore Roosevelt?

We can look at this:

As already stated, it is not known which individual actually coined the proverbial slogan. It probably was not Sheridan, and it also was not an even more famous, or rather infamous, Indian fighter who made the following remarks at a speech in January of 1886 in New York

"I suppose I should be ashamed to say that I take the Western view of the Indian. I don't far as to think that the only good Indians are dead Indians, but I believe nine out of every ten are and I shouldn't like to inquire too closely into the case of the tenth. The most vicious cowboy has more moral principle than the average Indian. Turn three hundred low families of New York into New Jersey, support them for fifty years in vicious idleness, and you will have some idea of what the Indians are. Reckless, revengeful, fiendishly cruel, they rob and murder, not the cowboys, who can take care of themselves, but the defenseless, lone settlers on the plains. As for the soldiers, an Indian chief once asked Sheridan for a cannon. 'What! Do you want to kill my soldiers with it?' asked the general. 'No,' replied the chief, 'want to kill the cowboy; kill soldier with a club.' "4 [As quoted in Hagedorn 1921:355]

The person who said these words was that "rough rider" who published his racist and expansionist views and an account of his exploits on the American frontier in his acclaimed book The Winning of the West (1889)-Theodor Roosevelt (1858–1919), who became President of the United States only five years after making these hateful comments. The fact that Roosevelt include proverb in a speech in 1886 in the eastern city of New York, far removed from the racial strife at the frontier, is a clear indication that the proverb and its discriminatory message had permeated the American consciousness.
_{–– Wolfgang Mieder: "'The Only Good Indian Is a Dead Indian': History and Meaning of a Proverbial Stereotype", The Journal of American Folklore, Vol. 106, No. 419 (Winter, 1993), pp. 38-60.}

Note that Roosevelt was quite prejudiced indeed towards Indians but the exact speech above is only quotable indirectly, through Hagedorn. It might be too early to jump to the conclusion that later compilers were not as "permeated by that sentiment" so that they omitted this part. Fact is that this proverb that was also ascribed to Sheridan gives a hint at the darker sentiments at the time.

Summary

No. While the words "final solution" do appear in this context, they don't mean the same thing as "the final solution". The quoted passage is about integrating people into society, not killing them. And while some people may have thought of just killing off the remaining Indians in grandiose words full of contemporary racism, they actually didn't set out to really kill everyone they could spot. To combine the word pair with the existing racist attitudes in that society stretches any artistic license a bit too much.