Yes there is a federal law against it. Ross Perot was going to do the same thing in the ‘70s. George H. W. Bush, then head of the CIA told him of the law. That’s why Perot hated Bush and entered the ‘92 presidential race to split the vote.
Perhaps he will help fund this military endeavor.
While the US may prohibit the use of mercenaries, there is little likelyhood of prosecution by the US. Of course, that would be the least of your worries in this mess—the legal systems of the Middle East are rather unpredictable, to put it mildly.