The proper way to go about it would be to compute the energy your pellets are carrying at the range you intend to fire over. You should be looking for around 9 ft lbs minimum on target for crow and rabbits.Here's a more practical test, fill one of those 3 lb metal coffee cans with water and place it at your chosen distance, marking a 1" diameter circle on it. If can you consistently land all your shots in that circle at your chosen range and the pellet has enough energy to go clean through the can when full of water, you should be good to go.
A stock 1377 wont be good for much except maybe a very close, excellently placed shot on very small game, mice, maybe a small squirrel.Still the chances of wounding rather than killing them outright are pretty high.
Like I said, at close range with a well placed shot.Personally I do not have enough confidence in my skill set to even try it Now put the Bandit in my hand and that's another story