Many years ago, when I first started shooting the LCP, I really did not like it. And alway's had my finger hurt with banging. Over time with so many rounds down range, I had no issue. No idea why I can think of other than just got use to the gun. For myself, I find the LCP's a much harsher shooting gun than both the Kahr and Pico both of which I find very mild shooting. I do know that the LCP's at least the one's I owned required a much tighter grip. All my LCP's were reliable, never had any issues with light strikes. The APX is a unique bullet. I never used them other than my P32 Keltec and they work fine, however I have read so many times they do not feed in a lot of guns. I shoot the PrecisionOne Plus P for all my 380's for defense. Sounds like your LCP just does not like that round. Personally I would just stick to stock trigger and get use to it. Adding a lighter trigger spring never made much sense to me. Too much chance for a light strike.
I bought a new Pico once. When I got it, it was the 2nd Gen trigger which my others have, but this trigger for some reason was lighter. It shot well, but I had a couple of light primer strikes. I sent it back, told them the trigger was too light and it was returned with a heavier trigger and never any more light strikes. Might want to ditch the APX ammo.
I personally do not care for the square triggers, some love them, just not me. The Kahr Trigger IMO is the best trigger on the planet for self defense. But that is only my opinion. Good luck, keep at it, I am sure you will find a happy medium.