wow! thank you for bringing that down.
it could be that there will be a king before the bais hamikdash is built but it doesn’t have to be mashiach. i’ll explain:
we know that the yidden were told three mitzvos- to make a king,to wipe out amalek, to build the bais hamikdash. in that order. in fact by the first bais hamikdash shaul was king then he fought amalek then david prepared and shlomo built the bais hamikdash. by the second bais hamikdash zerubavel was like a king then they killed 75000+ amalekim by the story of purim then they built the bais hamikdash (agav they were bringing korbanos already before the bais hamikdash was built).
so too by the third bais hamikdash there will be a king then a fight against amalek then the bais hamikdash.
But the king doesn’t have to be mashiach.