The purpose of the council quoting Amos was to show that Paul's ministry to the Gentiles was not contrary to the prophets.
Paul's ministry, however, was not the fulfilling of the Amos passage.
James quoted from the Greek version of the OT (the Septuigant), but with a slight modification. Here are the verses from the Septuigant :
"In that day will I raise up again the tabernacle of David that is fallen down; and I will build again the ruins thereof, and I will set it up...That the residue of men might seek after the Lord, and all the Gentiles, upon whom my name is called, saith the Lord, who does all these things." (Amos 9:11-12; LXX).
James replaced the words "in that day" with the following words:
"After this I will return, and will build again the tabernacle of David, which is fallen down; and I will build again the ruins thereof, and I will set it up: That the residue of men might seek after the Lord, and all the Gentiles, upon whom my name is called, saith the Lord, who doeth all these things" (Acts 15:16).
James believed that this prophecy of Amos will not be fulfilled until after the Lord Jesus returns.
Correct!
Isa 16:5 And in mercy shall the throne be established: and he shall sit upon it in truth in the tabernacle of David, judging, and seeking judgment, and hasting righteousness.
Jer 23:5 Behold, the days come, saith the LORD[YHVH], that I will raise unto David a righteous Branch, and a King shall reign and prosper, and shall execute judgment and justice in the earth.
Jer 23:6 In his days Judah shall be saved, and Israel shall dwell safely: and this is his name whereby he[the Branch] shall be called, THE LORD[YHVH] OUR RIGHTEOUSNESS.
Zec 6:12 And speak unto him, saying, Thus speaketh the LORD of hosts, saying, Behold the man whose name is The BRANCH; and he shall grow up out of his place, and he shall build the temple of the LORD:
Zec 6:13 Even he shall build the temple of the LORD; and he shall bear the glory, and shall sit and rule upon his throne; and he shall be a priest upon his throne: and the counsel of peace shall be between them both.