I use L&R replacement on two of my T/C Hawken Caplocks, if they are any better, in any way, than the original T/C locks, I am not astute enough to see it, I use them soley because they seem to work better with Davis DDST Triggers, which I do prefer 10x over the original T/C triggers.
I have owned several Lyman Rifles, both cap and flint, and I've never experienced any trouble, whatsoever certainly nothing like you have described.
Sorry I'm not much help, but it does sound to me that you may benefit from a replacement lock.
I have spent a lot of money over the years by having local "experts" repair my locks, and from my own silly mistakes trying to do it myself.
My suggestion is, if your lock ain't working, you just may be ahead of the game by getting a replacement.