Are we talking about part #22 in the diagram below? I thought the shift drum is held in place by the neutral switch (25).

mystic_1
In real life, the neutral shift plunger doubles as the detent for the shift drum. It is possible to pull the drum out a little if the parts on the side case are not all tight. This lets the forks misalign and the gear dogs drop out of their holes.
The case damage introduces another possibility: if the chain pushed far enough in, it could have hit and bent a shift fork. This could also cause the gears to not fully engage their dogs. The only cure there, though, is to pull the cases apart, and replace the fork(s) and possibly the cases (an expensive proposition!). Since the original pieces are gone, it would be difficult to make new fill-in plates there. I'm afraid this is a nasty situation...

A spare engine might be the better bet? They can often be had for $100-$200, it seems.