I think you boys are looking at the wrong spring as being the problem.
If the gun will fire using the unset trigger, then the main spring has enough ommf to do the job. from the photo we can see that this is a very rudimentary lock set-up and I think the probem is in the set trigger not slapping the prong on the sear hard enough that it wont clear the half-#### notch full and that is slowing the hammer drop sufficiently to not fire the cap. On all these inexpensive muzzle guns with set triggers , when the front trigger is set and that trigger is pulled it just releases the back trigger that has irs own spring and it "slaps" the prong in the photo.
I think, in this case that the small spring that engages the sear to the tumbler is to strong for the slap of the back trigger to overcome...two ways to remedy this....lessen the small spring tension on the sear a bit or strengthen the flat spring tension on the back trigger so the back trigger bar slaps harder.
You could prove or disprove my theory by removing the small sear spring and manually setting the sear in a fully cocked state and then re-assemble the gun, now try to fire a cap using the set trigger, if it fires you know the spring drag has been removed and where the problem actually is.