I do not have a LCP but I do have a Kel-tec Pta3. Field strip and clean the chamber really well with a brush. A little build up in the chamber can cause failure to go into battery. Lightly polish the feed ramp and a little molly grease on the rails. My LCP is not very fond Blazer aluminum cased ammo. It is the hardball fmj type profile. Polishing the feed ramp helped quite a bit. It does feed my reloads with the Hornady Truncated cone like XTP's without a hitch. Very good accuracy as well.
Is the ammo your shooting the same as you have been using before? I would say try different flavors of ammo but that is easier said than done. If you have another mag try it. A lot of my failure to feeds on other guns have been mag related. A recoil spring is another possibility but it should not be the case after only 500 rounds.