I have owned a number of pocket .380s but the only ones I have kept are my two LCPs, one a modified first generation and the other an LCP Custom.
I kept these two mainly because they were reliable and the others I owned (including various Kel-tecs, two SIG P 238s, an abysmal Kahr P380, an early Smith Bodyguard and an early Glock 42) were not.
The two problems with my first generation LCP were the complete lack of sights and the heavy trigger. Fixed both of these at some expense, but it looks to me like the second generation of the basic LCP has fixed both these issues, and I would get that over the LCP II, which by all accounts has a very light trigger requiring much care in handling and holstering.
The LCPs have fairly sharp recoil because they are so small and light. You can ameliorate this issue significantly by getting one of the Hogue wraparound grips that attach using a longer pin replacing the rear frame pin of the pistol. Helps a lot, and really, you should be able to deal with the recoil pretty well with that modification, and using the extended mag baseplates on your mags.
My Smith Bodyguard, like I said was an early version, and it was crap. Smith’s vaunted fast service couldn’t fix it. Maybe this gun is better now, in terms of reliability, but I would make sure of this before getting one. It’s other major issue has been a Very heavy trigger, which you can fix at additional expense by getting an Apex trigger kit put in. I wanted to like the Bodyguard, it’s comfortable to shoot compared to the LCP, and has good sights, but mine just wasn’t reliable.
I sold my unreliable Glock 42 to a buddy (with disclosure) who is a Glock armorer. He had problems too, but sent it back to Glock and they put in new parts which fixed his issues, so that gun may be OK now. It shoots much better than the LCP, but requires a full holster because of its trigger setup. A holster is really a requirement for pocket carry anyway.
People keep mentioning those SIG P238s, but as mentioned above, it’s a single action gun you should carry cocked and locked. You want to do that, OK. I don’t. Mine were also too finicky to be trusted for defensive carry. They shot nice.