Pushing forward on the trigger is supposed to "set" it so it fires under the lightest pressure. Proper behavior for this trigger is that (1) when not "set", pulling it back should release the cock with some minimal travel and nominal pressure to fire; (2) pushing forward should result in a click and the trigger entering "set" state (with no effect on the cock), and then (3) after it's "set" barely touching the trigger should release the cock to fire. If this is what it does
when it's disassembled and removed from the stock (disregarding the part about the cock) then your problem is most likely the trigger group isn't properly positioned with respect to the lock.
I'm guessing, and just guessing, based on your description and on my own experience troubleshooting a
double-set trigger (ie different mechanism), that your trigger group is probably
too close to the lock. Some would suggest filing some off the top of the trigger bar, but that's a permanent solution, so don't do it unless you're sure that's what the problem is.
I'd suggest troubleshooting by shimming the trigger group with a couple small washers under the trigger plate to move it farther away from the lock, then reassembling and see if your problem is solved. If that's the case, then filing
just the end of the bar that moves up when the trigger is pushed
forward might be the appropriate fix.
And I'm hoping, if I'm wrong, someone with more experience on
single-set triggers will jump in here and correct me. So, just have patience and let's see what develops.