I'm not even sure the president couldn't fire a special prosecutor for any old reason. Constitutionally, it's supposed to be a gentlemen's agreement, but technically it's under the executive branch, by constitutional design, and the president can fire that person. The constitution does not authorize the legislative branch to interfere with a core executive branch power.
Probably would roll out like this:
President fires the new-and-improved-unfirable special prosecutor anyway.
Congress: "He can't fire this executive-branch underling!"
Pres: "Yes, I can. I am the head of the executive branch, and I enforce the laws."
Supreme Court: "He's right."
Congress: "Fine, let's start impeachment proceedings."
Prest: "I resign effective 3:33 AM tomorrow morning, after one last tweet."