Hi Rolf,
Yes, you can weld metal to the top of the trigger and then case harden it with no problem. I have done that several times. However, before messing with the trigger, take a careful look at the lock. Watch how the sear bar moves up and down as you cock the lock from rest to half cock and then to full cock. In many cases the sear bar drops a little bit lower at full cock than when at rest. What usually happens is that you try to keep the trigger snug when the gun is uncocked but when you do that, the lock won't hold at full cock. So next you file down the trigger bar until the lock holds at full cock, only now there is slop when the lock is at rest. The fix is this: adjust the trigger bar so the lock cocks properly at full cock. Then anneal the tumbler and carefully grind a shallow depression (more of a shallow groove across the bottom of the tumbler really) on the bottom of the tumbler where the nose of the sear rests when the lock is uncocked. Deepen the groove sufficiently to eliminate any slop in the trigger when the lock is at rest. Reharden the tumbler and case harden the trigger and you should be all set to go. If you don't want to anneal the tumbler, a Dremel tool with a diamond grinding bit will work fine.
dave