It probably would help if you show where the Dome of the Rock is and how the Temple can still be built with that Dome of the Rock being in the outer courts of the Gentiles by the discovery of the foundation of one side of the wall of the former temple.
At least that was what one site indicated, but did not refer to the other holy building at that site.
Not sure I can find it again. it is possible that it was removed for fear of reprisals of those that would oppose its construction, regardless of it being done without removing the Dome of the Rock in the outer courts of the gentiles of that Third Temple.
What was the Court of the Gentiles in the Jewish temple?
It may be apparent after the rapture event when one third of the earth gets burned up and USA and the entire western hemisphere is gone for why the EU and the world will make a peace treaty to assure Israel that there will be no more attacks by allowing them to build that Third Temple.
Revelation 11:1And there was given me a reed like unto a rod: and the angel stood, saying, Rise, and measure the temple of God, and the altar, and them that worship therein.
2 But the court which is without the temple leave out, and measure it not; for it is given unto the Gentiles: and the holy city shall they tread under foot forty and two months.
3 And I will give power unto my two witnesses, and they shall prophesy a thousand two hundred and threescore days, clothed in sackcloth.