Belichick carried 4 QBs the year they drafted Brady (Bledsoe was starter and they had Friesz and Bishop also). But when it comes to camp there are no issues with bringing in 4 or even 5 QBs and then cutting the two who lose out.
Mannion needs to compete straight up with Keenum and Foles, assuming all are still on the roster. The loser gets cut, competition is good for the team. If the Rams bring in another QB then it's a four way battle and cut the guy who is least sharp early in the process due to snap availability or trade Foles to another team.
The rookie of course gets the same ride this year that Mannion just got. Assuming they draft one of course, which I do think we can assume. But Mannion will not be competing with that rookie, he'll be lined up against the vets this go round.
I expect the Rams to carry 3 QBs, either way. Starter, backup, future. Unless they trade up for Goff, in which case I suspect he'd win the backup role rather easily.