I went back to that range and fired both of them, for a total of 100 rounds through each.
Hellcat shot better/more accurate, but my trigger finger dragged on the guard and I have small hands. It was annoying. The trigger requires more force, but brakes cleaner than the 365 which has takeup and then a mushy squeeze. Hellcat was more crisp. Hellcat had better ergo, with the stippling up front. Hellcat slide failed the stay back 3 times the second day I shot it. Hellcat has a snappier recoil impulse.
Sig 365 had better/lighter recoil impulse, more generous trigger guard, and I think the trigger is technically better - well maybe not. In any case I could not hold as tight groups with it as the above ones shot with the hellcat.
In the end, the experience helped point out each guns weaknesses and I decided screw it, I'll keep my S&W M&P shield until, inevitably, other models are released from other manufacturers and I can try them.