How do members of the Church of Jesus Christ of Latter-Day Saints reconcile an unchangeable God with the concept of eternal progression?

Fear not, one of the institute manuals has you covered, the question is not as esoteric as you think.

The teachers manual for the course "Gospel doctrine" from 1987 and 2000 has this:

What does the King Follett discourse teach about the nature of God? Does it teach >that God continues to progress throughout the eternities? If so, how does God >progress? Ask the students the following questions:

Does God progress in attributes and characteristics? (No. He is perfect in these >things. See Matthew 5:48; Alma 7:20.)

Does God progress in knowledge, light, and truth? (No. He has a fulness of >knowledge, light, and truth. See D&C 66:12; 2 Nephi 2:24.)

Does God progress in power or in his ability to accomplish his work? (No. He has >all power, though he will not violate eternal law nor the agency of man. See Alma >26:35; Luke 1:37; 1 Nephi 7:12; Mosiah 4:9.)

Yet God does progress. To explore the nature of God’s progression, read the >statement by the Prophet Joseph Smith on page 8 of the student manual (see >Teachings, pp. 347–48).

And the student manual cites Joseph Smith like this:

“What did Jesus do? Why; I do the things I saw my Father do when worlds came rolling into existence. My Father worked out his kingdom with fear and trembling, and I must do the same; and when I get my kingdom, I shall present it to my Father, so that he may obtain kingdom upon kingdom, and it will exalt him in glory. He will then take a higher exaltation, and I will take his place, and thereby become exalted myself. So that Jesus treads in the tracks of his Father, and inherits what God did before; and God is thus glorified and exalted in the salvation and exaltation of all his children” (Smith, Teachings, 347–48).

Personally, I think it's awkwardly expressed, but in my own words: Since God's work and glory is to "bring to pass the immortality and eternal life of man"(Moses 1:39), that is how / in what area he progresses. And that is also the meaning of "eternal progression" for us, not that we will only approach but never achieve the level of perfection the Father has. That is not an unattainable goal, but one of the waypoints on the path. See also D&C 131.

Teachings, 347-348:

Perhaps no passage in the Prophet's discourse has given more offense than the one here noted, and yet men are coming to think and feel the truth of what he said. Henry Drummond, for instance (following the Prophet by half a century), in his really great work, Natural Law in the Spiritual World, in the chapter on Growth, wherein he points out the difference between the merely moral man and one whose life has been touched by the spiritual power of God, and so received something that the merely moral man has not received, says: "The end of salvation is perfection, the Christ-like mind, character and life. ... Therefore the man who has within himself this great formative agent, Life [spiritual life] is nearer the end than the man who has morality alone. The latter can never reach perfection, the former must. For the life must develop out according to its type; and being a germ of the Christ-life, it must unfold into a Christ." Joseph Smith's doctrine means no more than this.

Sir Oliver Lodge says much to the same effect in the following passage on "Christianity and Science" (Hibbert's Journal, April, 1906):

It is orthodox, therefore, to maintain that Christ's birth was miraculous and his death portentous, that he continued in existence otherwise than as we men continue, that his very body rose and ascended into heaven-whatever that collection of words may mean. But I suggest that such an attempt at exceptional glorification of his body is a pious heresy-a heresy which misses the truth lying open to our eyes. His humanity is to be recognized as real and ordinary and thorough and complete; not in middle life alone; but at birth, and at death and after death. Whatever happened to him may happen to any one of us, provided we attain the appropriate altitude; an altitude which, whether within our individual reach or not, is assuredly within reach of humanity. That is what he urged again and again. "Be born again." "Be ye perfect." "Ye are the sons of God." "My Father and your Father, my God and your God." The uniqueness of the ordinary humanity of Christ is the first and patent truth, masked only by well-meaning and reverent superstition. But the second truth is greater than that-without it the first would be meaningless and useless,-if man alone, what gain have we? The world is full of men. What the world wants is a God. Behold the God! - [That is, the God, Jesus Christ.]

The divinity of Jesus is the truth which now requires to be reperceived, to be illumined afresh by new knowledge, to be cleansed and revivified by the wholesome flood of skepticism which has poured over it: it can be freed now from all trace of groveling superstition: and can be recognized freely and enthusiastically: the divinity of Jesus, and [the divinity] of all other noble and saintly souls, insofar as they, too, have been inflamed by a spark of Deity-insofar as they, too, can be recognized as manifestations of the Divine.-Notes by Elder B.H. Roberts