I know you said you can't use a .22, but what about a pellet gun? Some of those break action ones are about the same power of a .22.
I wonder if a pellet done would get it done. I think they have very thick hides.
The main issue that I have with them besides not wanting them living in my barn is that they get into my FENCED garden. It took me a while to figure it out, but the fencing is 2"x4" and the juvenile groundhogs can fit their heads through and crawl right through.
By the way, my best ground hog story is the time I got home and found one in my fenced garden. I took off full speed towards the garden. He saw me and went to squeeze back through the fence at the back of the garden. He popped through right as I made it around back. So it was basically pinned with me in front of him and the fence behind. I took a couple steps back and picked up a big rock. When I went for the rock he made his move. I managed to anticipate his speed and direction and landed the rock right on top of him (it was a small boulder so more of a heave than a throw). Took one more bigger rock to finish the job. I was surprised at the adrenaline rush from the whole encounter. Took a glass of bourbon to level me out.