As for the building of the tabernacles for Moses, Elijah and Christ, I think that Moses and Elijah could be the two that sit on Christ's right and left hand in the millennium (Mark 10:40),
And the tabernacle is not but a fulfillment of OT prophecy (Ezekiel 37:27).
Moses' and Elijah's tabernacles could be a fulfillment of Hosea 12:9:
KJV
And I that am the LORD thy God from the land of Egypt will yet make thee to dwell in tabernacles, as in the days of the solemn feast.
NASB
But I have been the Lord your God since the land of Egypt;
I will make you live in tents again,
As in the days of the appointed festival.
It also was not uncommon for the Jews to dwell in tents as opposed to households:
Jeremiah 35 NASB, bold emphases mine
7 You shall not build a house, and you shall not sow seed and you shall not plant a vineyard or own one; but in tents you shall dwell all your days, that you may live many days in the land where you sojourn.’ 8 We have obeyed the voice of Jonadab the son of Rechab, our father, in all that he commanded us, not to drink wine all our days, we, our wives, our sons or our daughters, 9 nor to build ourselves houses to dwell in; and we do not have vineyard or field or seed. 10 We have only dwelt in tents, and have obeyed and have done according to all that Jonadab our father commanded us. 11 But when Nebuchadnezzar king of Babylon came up against the land, we said, ‘Come and let us go to Jerusalem before the army of the Chaldeans and before the army of the Arameans.’ So we have dwelt in Jerusalem.”
12 Then the word of the Lord came to Jeremiah, saying, 13 “Thus says the Lord of hosts, the God of Israel, ‘Go and say to the men of Judah and the inhabitants of Jerusalem, “Will you not receive instruction by listening to My words?” declares the Lord. 14 “The words of Jonadab the son of Rechab, which he commanded his sons not to drink wine, are observed. So they do not drink wine to this day, for they have obeyed their father’s command. But I have spoken to you again and again; yet you have not listened to Me. 15 Also I have sent to you all My servants the prophets, sending them again and again, saying: ‘Turn now every man from his evil way and amend your deeds, and do not go after other gods to worship them. Then you will dwell in the land which I have given to you and to your forefathers; but you have not inclined your ear or listened to Me. 16 Indeed, the sons of Jonadab the son of Rechab have observed the command of their father which he commanded them, but this people has not listened to Me.’”’ 17 Therefore thus says the Lord, the God of hosts, the God of Israel, ‘Behold, I am bringing on Judah and on all the inhabitants of Jerusalem all the disaster that I have pronounced against them; because I spoke to them but they did not listen, and I have called them but they did not answer.’”
18 Then Jeremiah said to the house of the Rechabites, “Thus says the Lord of hosts, the God of Israel, ‘Because you have obeyed the command of Jonadab your father, kept all his commands and done according to all that he commanded you; 19 therefore thus says the Lord of hosts, the God of Israel, “Jonadab the son of Rechab shall not lack a man to stand before Me always.”’”
Ryrie says, about Hosea 12:9 above, that:
Israel would once again live in tabernacles (tents), because her cities would be destroyed. The Festival of Tabernacles commemorated Israel's wilderness years after they left Egypt. Their coming exile would be another such wilderness experience.
Ryrie, C. C. (1994). The Ryrie KJV Study Bible. Chicago: Moody Bible Institute. p. 1303.
Maybe the Jews will live in tents while the Millennial temple is being built.
Maybe the transfiguration shows us that Christ, who is able to subdue all things unto himself, will build the temple without opposition (as contrasted to Nehemiah - Dan. 9:25). In Ezekiel, we read that "they" will build the altar (43:18). It doesn't specify
who will build it. I think that Christ will coerce the Jews to build the temple, and the transfiguration gives us a glimpse of the power of Christ that will be manifest in the millennial by His coercion of Peter's mouth. "Master, it is good for us to be here: and let us make three tabernacles; one for thee, and one for Moses, and one for Elias: not knowing what he said" (Luke 9:33, KJV), "for they were sore afraid" (Mark 9:6, KJV).