Okay, overdue UPDATE!
I've been out of town for 2 weeks, but the Renegade is back in our possession and all is good!
Would you believe me if I told you it was a fuse?
It was still having starting issues after they replaced the brake switch.
Turns out the starter relay fuse had a partial break on one side (bottom left).
View attachment 484728
So while it wasn't broken enough to prevent a good current flow test or for it to start "sometimes", it was enough of a break to not start other times. Could have been temperature dependent, or it could have just been entirely random when current was able to flow through. Either way, it's been starting fine the last 2 days so that seems to have fixed it!
I'll have to remember to actually perform a good physical inspection of fuses next time rather than relying entirely on a meter.
I really appreciate everyone's comments and suggestions! Y'all have a lot of good knowledge here.