Jumbo... I have both the P3AT and LCP... my P3AT had a problem where the axis pin located behind the trigger had slipped down under recoil... and in doing so, came loose from the top of the trigger axis which connects the polymer trigger to the draw bar that engages the hammer... as it was, the trigger could rotate enough to contact the frame and not fire the pistol... mine was an old one without the external extractor spring, and this may have been corrected in the newer models... the LCP has a shouldered pin and cannot slip down...
my LCP shoots a little low, so I added a set of rear sights... I am really contemplating adding a front sight, but there is very little room to dovetail... possibly silver solder a sight base
I would opt for the LCP... the new S@W Bodyguard without the laser might be an option, but IIRC is slightly larger than the LCP...
with the small sight radius, and longer trigger pull of the pocket .380s, a front sight to watch is an option it seems has escaped the engineers...
on the next order of magnitude, the Glock 42 has a good trigger and the best set of Glock sights I've seen
rear sight
since retirement 6 years ago, I carry daily in an Andrews Custom Leather pistol pouch (upper left) with a kydex flap I've added to help keep from printing in nylon fishing shorts