Oh, absolutely Tywin marries Rhaegar to Cersei. In fact, if Rhaegar doesn’t, I think he suddenly cuts his head off while shaving, Cersei is betrothed to Viserys, and All Hail Regent and Lord Protector Hand of the King Tywin Lannister.
And at that point, I think the Southron Alliance changes targets to deal with the Kingslayer Tyrant (in either scenario, really, although the propaganda works better if Rhaegar had died), and picks up the Tyrells to boot, because I think with that much power the crab-bucket nature of feudal politics really begin to kick in.
I do not know if Tywin would kill Rhaegar so easily. I mean he did not seem to do anything against Aerys either and he had a reason to do so. Tywin is ambitions in my opinion but not so power-hungry he would be that extreme.
He’d have a pretty overwhelming reason – if Rhaegar refuses Tywin, there’s a good chance Rhaegar sees Tywin as a traitor and kingslayer.