How do functionalists explain the fact that Holocaust continued until the very end of the war?

In the BBC's 1997 documentary 'The Nazis: A Warning From History', the rise and fall of the Nazi regime is examined. Another insightful analysis is Timothy Snyder's 2015 book 'Black Earth: the Holocaust as History and Warning'.

From these sources we gain an important insight into the character of the Nazi regime, and how fundamentally different their thinking was. This cannot be stressed enough, that you cannot merely think of the Nazis and Hitler as just nationalists, or just antisemites.

Their behaviour contrasts with the behaviour of antisemitic nationalists, like Antonescu's Romania. Romanians helped to exterminate 300,000 Jews... and then completely reversed their policy, because Antonescu's primary objective was national supremacy, and not racial supremacy inclusive of a life or death struggle against international Jewry.

The Nazis, and Hitler in particular, were racial anarchists. Snyder goes into this in detail, but the gist is this: Hitler believed that the natural way of things is violent struggle between races, allowing the superior to be superior. Hitler also believed that the legal and philosophical evolution of civilisation, from Christianity to ethics and law, were created and maintained by an elaborate global Jewish conspiracy to alienate people from their natural struggle. To enable allegedly inferior Jews to keep allegedly superior Germans under control, and pervert the natural order of things.

Consider how the popular German response to their losing the First World War was a 'stabbed in the back' myth. Allegedly, it was Jews that were responsible for selling out, and if only they had continued fighting they would have achieved victory (which was completely fanciful thinking). The Germans then had to suffer the treaty of Versailles, again blamed on Jewish elements.

Of course it doesn't make sense to continue a genocide when those resources could otherwise be allocated to a war effort, especially when the war is going badly. But that rational response is a distinctly un-Nazi one.

To the Nazis fighting in the east, there was little distinction between efforts to exterminate Jews, and fight Russians. They sincerely believed that the liberal regimes in Britain and America, along with the Communist regimes in the USSR, were two sides of the same Jewish coin, and were both ultimately controlled by a giant Jewish conspiracy.

Furthermore, it is important to add that the Nazis had become complacent and optimistic after swift conquests in Poland and France. The Nazis believed sincerely that an attack on the Soviet Union would result in another swift victory. They perceived the USSR's apparent strength to be merely a facade, part of an illusory system of Jewish conspiracy and control. But when the superior German people were sent to knock the facade down, the whole house of Judeo-Bolshevism would collapse. This idea initially seemed correct as the surprised and unprepared Soviet army struggled to cope. But as the war turned against them the delusion proved problematic, which for most committed Nazis served only to increase their denial and zealotry.

Most Nazis regarded the war, especially against the Soviets, not simply as a war between nations, but as a war of annihilation to be fought to the bitter end. Emboldening this fervour was a sincere belief in German racial supremacy, and implicit in this is the belief that they could not lose against a racially inferior foe.

To conclude, as Snyder said in an interview on the subject:

At the end of the war, Hitler said, ‘Well the Germans lost, that just shows the Russians are stronger. So be it. That’s the verdict of nature.’ I don’t think a nationalist would say that.

As the end of the war was approaching, we see more and more effort on the part of at least the people in the field towards eliminating evidence of their actions. This led to prisoners (both Jews, other concentration camp inmates, and prisoners of war) being transported (often on foot for lack of trains and trucks, as well as to kill them through exhaustion) towards camps ever nearer the German heartland. At the same time documents were trucked there as well, or sometimes hidden or destroyed (rare) on site, and where possible facilities destroyed as well.
The "deathcamps" no longer existed at the time (meaning, the special facilities set up for no other purpose than to gas people, as far as I know those were all shut down by mid-late 1944 at the latest, as it was decided to work Jews to death instead, using them as slave labour in the production of e.g. the V2 and Me262).

I've no expectation (or intention) of supplanting the accepted answer, but I would like to add some additional information.

For a start, nobody is a pure functionalist nor a pure intentionalist. Even Goldhagen, who you mention a few times in your question, acknowledges the extent to which the Nazis became progressively more radicalised as the war progressed, and even Goldhagen doesn't suppose that the Nazi party adopted a broad program of physical annihilation prior to 1941.

To complicate this picture, consider that the intentionalism/functionalism dialectic operates both in relation to the NSDAP as a party, and to individual members of that party as people. Even if we were to suppose that the leader of the party had in mind a program of genocide before the war started, that is insufficient to demonstrate that the party itself was structured in such a way as to expedite the fulfillment of that goal.

To speak directly to the substance of your question: how do "functionalists" (bearing in mind that there is also no such thing as pure functionalism) explain the desperate attempts made at the conclusion of the war to continue murdering large numbers of people?

For a start, the widespread murder of people with physical and intellectual disabilities continued in German hospitals well after their occupation by the Allied powers. This is because the doctors responsible for this program were genuinely convinced of its intrinsic goodness. They did not associate euthanasia with murder.

Secondly, the murder of Jews in the final stages of the war occurred concurrently with the murder of other "undesirables", and it was not just Jews who were marched back from the approaching front with staggering casualties.

Thirdly, the desire to murder prisoners at this late stage corresponds with other procedures that the Nazis and their collaborators implemented in order to limit the amount of physical evidence that their murderous campaign had generated.

Sonderaktion 1005 was the code-name given to a widespread program of disinterment, the burning of corpses and the shredding of documents. During this program, death camps at Sobibor, Belzec and Treblinka was dismantled and removed, and the teams of slaves employed in this program were subsequently executed.

Finally, when it became more apparent even to the die-hard Nazi ideologues that the war was over, it was still not necessarily apparent the extent to which they had lost. Many people believed that while they might not be able to hold onto their empire, they will still be able to retain control of Germany, and that they will only be able to do this if they are in a sufficiently strong condition as to sue for peace.

Forcing the Hungarians to participate in the slaughter of their Jewish population was not at all an irrational decision. The Hungarian government (which went through six prime ministers in the time of the war) consistently prevaricated on this issue, and it was clear to observers in Berlin that they were going to join the Allies. Making them complicit in the final solution was designed to prevent that possibility, and to cause them to fight alongside Germany with greater desperation.

The idea that this late campaign damaged the war effort is mistaken, and it is an historiographical error that is only now starting to be corrected. In all instances, "human cargo" destined for Auschwitz took second place to the transport of munitions and soldiers, which is why cattle trains often languished for days on end on railway spurs.