If I snapped the head off of a 1/4" bolt with the correct TPI trying to get it in the hole, I'll do the same with a 1/4" self-tapper. The issue is that a 1/4" shank is too big.
I like the idea, but I'm not seeing 7/32" self-tappers. Be willing to try, though, if you have a link.
To be honest, I will wager there's something else going on here. I highly doubt broke-*** Chrysler would blow coin on a weird limited-application screw when a 1/4-20 would work.