I did a quick search in the LXX and it landed me in the book of Acts where James seems to shed light on the passage.
“After they finished speaking, James replied, "Brothers, listen to me. Simeon has related how God first visited the Gentiles, to take from them a people for his name. And with this the words of the prophets agree, just as it is written, "'After this I will return, and I will rebuild the tent of David that has fallen; I will rebuild its ruins, and I will restore it, that the remnant of mankind may seek the Lord, and all the Gentiles who are called by my name, says the Lord, who makes these things known from of old.' Therefore my judgment is that we should not trouble those of the Gentiles who turn to God,”
The booth is therefore God’s Israel.
“For neither circumcision counts for anything, nor uncircumcision, but a new creation. And as for all who walk by this rule, peace and mercy be upon them, and upon the Israel of God.”
This is tied into the fact that all of Israel, all twelve tribes will be saved which is a reflection of David’s kingdom, as he had all twelve tribes under his reign.
please check this link out about the twelve tribes
This is also consistent with the passage about the wheat and the chaff two verses earlier in Amos 9
“"For behold, I will command, and shake the house of Israel among all the nations as one shakes with a sieve, but no pebble shall fall to the earth.”
Notice it’s not the house of Judah which would have been just the house of David. Judah and Benjamin but it’s all Israel. And the fact that not even a pebble falls down after the shaking says that these grains, are the true house of Israel, in that day or after that day.
the relationship of wheat and chaff
It comes down to God bringing about the true Israel just as David was a man after God’s heart so will the Israel of God be after God’s own heart.
“But it is not as though the word of God has failed. For not all who are descended from Israel (by bloodline) belong to Israel (by the will of God John 1:13),
This means that it is not the children of the flesh who are the children of God, but the children of the promise are counted as offspring.”
Romans 9:6, 8
That is my understanding of what the booth of David refers to, the restored twelve tribes not according to bloodlines but by faith in Jesus Christ.
In light of the clarification to the question, let’s explore the choice of wording for booth. As opposed to tabernacle, tent or house.
Fact is that most people in Israel still lived in tents even in the time of Rehoboam and much later in the times of the prophets.
1Ki 12:16 And when all Israel saw that the king did not listen to them, the people answered the king, “What portion do we have in David? We have no inheritance in the son of Jesse. To your tents, O Israel! Look now to your own house, David.” So Israel went to their tents
But it’s more than just a reference to a tent it’s a booth, sukkot after all, which should trigger the reference to the feast of the sukkot or tabernacles as it’s translated.
This is the last and final feast of the year when Israel would build booths to celebrate before the Lord the first fruits. Leviticus 23:33-36
It also says it’s a repairing of the existing booth not a new booth. This is referencing the house of David 1 Chronicles 17 which is mirrored in the angel’s announcement to Mary
“He will be great, and will be called the Son of the Highest; and the Lord God will give Him the throne of His father David. And He will reign over the house of Jacob forever, and of His kingdom there will be no end.””
This is saying that Amos 9:11 will happen when the kingdom of David will be re-established
It’s David’s booth because the king clearly had a part in the feasts of the sukkot.
It’s also a booth because it’s not all of Israel, it’s only the remnant, those few who truly love God, who are faithful.
“So the daughter of Zion is left as a booth in a vineyard, As a hut in a garden of cucumbers, As a besieged city. Unless the LORD of hosts Had left to us a very small remnant, We would have become like Sodom, We would have been made like Gomorrah.”
The booth and the small remnant are equated. This is mirrored in Jesus’ words
“Because narrow is the gate and difficult is the way which leads to life, and there are few who find it.”
But the remnant is also a Holy SEED; so booth, remnant, Holy Seed all the same
“But yet a tenth will be in it, And will return and be for consuming, As a terebinth tree or as an oak, Whose stump remains when it is cut down. So the holy seed shall be its stump.””
The booth draws on the sukkot feast which is the remnant that comes down the restored line of the Davidic throne, they represent God’s portion, which is the FIRST fruits or the tithe.
(I don’t want anyone to get the impression that just because few find it and percentage wise it’s a smaller number that they are numerically not many
““I will surely assemble all of you, O Jacob, I will surely gather the remnant of Israel; I will put them together like sheep of the fold, Like a flock in the midst of their pasture; They shall make a loud noise because of so many people.”