Wanted to provide an update. I did some research and found that when doing a function test you are supposed to hear two distinct audible clicks when the trigger is resetting. From the factory my LCP custom was not doing this. I could only hear one audible click and then if I smacked the top of the slide I could hear a second. Then the trigger would reset fully.
Ruger doesn't define a function test in the owners manual, but it can be defined as follows.
1) Make sure the gun is clear
2) Rack the slide to ensure the hammer is cocked
3) Pull the trigger and keep it held to the rear
4) Rack the slide again
5) Release the trigger. You should hear two audible clicks as it is reset
6) Pull the trigger again to verify it is working
Since I knew my trigger wasn't resetting properly I decided to take it further apart to examine if everything seemed to be assembled correctly based off the diagram in the owners manual. I removed the frame insert and everything appeared to be correctly assembled. So I put the gun back together and did the function test. Now the trigger is resetting properly and the gun is functioning as I would expect. I have no idea what was causing my original problem, maybe there was some extra polymer flakes binding something up. I also read somewhere that if the hammer catch in the frame isn't properly aligned into the hammer in the slide that it can also cause trigger problems, but I have no way to verify if that is correct. I will update once I can make it to the range for another test.