Crazy Horse, greetings from the GFN (Great Frozen North). Don't give up on this rifle, and don't give up on flinters in general.
Odd that your frizzen is soft; i had the exact same problem with my Thompson Center, and a 'smith said he could reharden my T/C frizzen for twenty bucks, but i really didn't want to do that. why not, i asked, and he reached under the counter and pulled out a Lyman frizzen and sold it to me for fifteen (this was a long time ago).
the advice about turning the flint over is worth a try. your flint should hit the frizzen about two thirds of the way up. try the search function- if i remember, Paul V. had a good post on how this is supposed to work. I won't get into lead vs. leather- except to urge you to try both and use what works best for you.
try drilling the touch hole liner to the next size. go slow, and i wouldn't go past 5/64. it doesn't hurt to have a spare, by the way.
i would try talking to Lyman again. See if they will be willing to send you another frizzen. If that doesn't work, try a replacement frizzen from Track of the Wolf.
As regards the replacement of the cock, i didn't know you could do that, but the photos just illuminate my ignorance. Probably more than you would like to spend on the problem, but cheaper that the alternative replacement lock, i'm pretty sure. (that replacement lock is pretty good looking, though).
set triggers can be problematical. having said that, i'd try taking the trigger out of the stock and see if it works any better. the slightest bit of pressure from a bit of wood chip in the inlet will trash an otherwise good trigger. absent that, the R.E. Davis is great- i've used them in all of my builds and they work fine. again, probably more than yo want to spend, but a good investment nonetheless.
good luck with your rifle, and please let us know how it turned out