James was not the one to supply that reading. He is not modifying the text and then claiming "it is written". His LXX must have read differently from the versions we have. If he changed it and asserted that he was simply quoting it and appealing to its authority then he is a fraud, no? It is possible though that he is conflating the verse with another related verse though I know not which off the top of my head.
Walvoord's point about him referencing a still future enthronement is spot on. The messiah is "the lord's anointed one" which refers to his being that designated and promised descendant of David to be king of Israel forever:
Luk_1:32 He shall be great, and shall be called the Son of the
Highest: and the Lord God shall give unto him the throne of his father
David:
Jesus' eternal destiny is to rule Israel forever as king:
Luk 1:33 And he shall reign over the house of Jacob for ever; and of
his kingdom there shall be no end.
It is only during this period called "the times of the gentiles/nations" that Jesus' reign has been temporarily extended so that he "rules in the midst of his enemies with a rod of iron":
Psa_2:9 Thou shalt break them with a rod of iron; thou shalt dash
them in pieces like a potter's vessel.
Rev_2:27 And he shall rule them with a rod of iron; as the vessels of
a potter shall they be broken to shivers: even as I received of my
Father.
Rev_12:5 And she brought forth a man child, who was to rule all
nations with a rod of iron: and her child was caught up unto God, and
to his throne.
Rev_19:15 And out of his mouth goeth a sharp sword, that with it he
should smite the nations: and he shall rule them with a rod of iron:
and he treadeth the winepress of the fierceness and wrath of Almighty
God.
Notice that God promises to temporarily "send out" the messiah's influence and rule unwilling peoples:
Psa 110:1 A Psalm of David. The LORD said unto my Lord, Sit thou at
my right hand, until I make thine enemies thy footstool. Psa
110:2 The LORD shall send the rod of thy strength out of Zion: rule
thou in the midst of thine enemies. Psa 110:3 Thy people shall
be willing in the day of thy power, in the beauties of holiness from
the womb of the morning: thou hast the dew of thy youth. Psa 110:4
The LORD hath sworn, and will not repent, Thou art a priest for ever
after the order of Melchizedek. Psa 110:5 The Lord at thy right hand
shall strike through kings in the day of his wrath. Psa 110:6 He
shall judge among the heathen, he shall fill the places with the dead
bodies; he shall wound the heads over many countries. Psa 110:7 He
shall drink of the brook in the way: therefore shall he lift up the
head.
However, once he has subdued the nations (via the harrowing curses and plagues in Revelation) he will rebuild the temple, restore the nation of Israel and Israel will become one of the subject nations and God himself will once again be KURIOS over all the world:
1Co 15:24 Then cometh the end, when he shall have delivered up the
kingdom to God, even the Father; when he shall have put down all rule
and all authority and power. 1Co 15:25 For he must reign, till he
hath put all enemies under his feet. 1Co 15:26 The last enemy that
shall be destroyed is death. 1Co 15:27 For he hath put all things
under his feet. But when he saith all things are put under him, it is
manifest that he is excepted, which did put all things under him. 1Co
15:28 And when all things shall be subdued unto him, then shall the
Son also himself be subject unto him that put all things under him,
that God may be all in all.
So James' more accurate LXX reveals that God's "turning away" from Israel is only temporary for God will "return" to the Israel program after the "Church Age" in which we find ourselves.
Now:
Messiah's rule is from God's right hand from which he serves as KURIOS as God places the hostile nations under his feet via demoralizing acts of divine terrorism.
When God "returns" to the Israel program:
The temple will be rebuilt and Jesus will rule Israel among a willing people (Psalm 110:3). God will rule the world over all the nations of the earth (including Israel) as king of kings (with Jesus being one of the subject kings).
KJV unless otherwise noted