There was a member here that had a Champion radiator that was stamped "374B" that had to modify his lower core support to get it to fit, but I haven't seen anyone else that's needed to do that, and there's quite a few people running the CC374 (although technically it's an E-body radiator).
Here's what I did:
I massaged the driver's corner of the lower radiator support with a dead blow hammer. I only did this because mine was actually damaged, it had a dent from hitting something. But I did do that, so maybe I added some clearance. Doubtful.