It doesn't necessarily sound like a problem with the fly to me. Check the screws, as described above. It would be a good idea to take the lock out of the stock and examine it. if the lock works out of the stock, it may be a stock problem. If you can isolate the problem to the lock, the problem could be a weak or broken sear spring that won't push the sear nose into position, or maybe the full-cock notch in the tumbler is worn, or the sear nose is worn. If it is a problem with sear/tumbler engagement, the sear nose and full cock notch may need stoning to improve their fit.
If the fly is missing and the rifle has been used repeatedly with set triggers anyway, the sear nose may have been damaged by hitting the half-cock notch in the tumbler as the hammer falls. If that is the case, you would need a new fly, but you would still need to fix the sear and likely the tumbler notches, anyway.
Brad or Shane Emig at Cabin Creek Muzzleloading could probably fix it. I have heard that Lee Shaver works on locks, and I have also heard that there is a gunsmith at the Log Cabin Shop in Ohio who can fix locks.
Good luck with it!
Notchy Bob