I do not see the "false Messiah" anywhere in the scripture. You are told from Daniel and Revelation that the beast shall rule for 3 and a half years that's all and has its roots in the Roman Empire. He shall strike a deal with many guaranteeing peace and maybe letting the Jews build a third temple sure. But my point stands - when the harlot church takes over many Jews shall realize that they either have to accept its religion or die. Many shall mourn for Christ then and repent in ashes. That is why the two witnesses are dressed in sackcloth for they know many Jewish people shall die.How can the false Messiah the Jews will accept to rule in the tradition of the Jews be able to do that with no Temple ?
His ability to make peace with all the enemies surrounding Israel & allow a Temple to be built will be a large part of what causes Him to be accepted as their long awaited Messiah.
Jhn 5:43
I am come in my Father's name, and ye receive me not: if another shall come in his own name, him ye will receive.
And you are forgetting about the King of the South or Islam. The king of the North in the end time shall defeat him and Islam shall cease to exist as a religion. That much I know. Now the religion of the harlot church shall not be Christianity either. It shall be a worship of the beast influenced by Satan. And in the end times this beast shall be bsy with Kings of the East too towards the end.