We owned a small ranch in SoCal for 15 years with a variety of barnyard animals. This was in a former walnut grove and at times the ground squirrels and rats would just get out of control. I got a RWS Diana in .177 and it was quite effective with this sized varmint. I probably bagged 50 rats and a couple dozen ground squirrels with that simple German air rifle.
For ground hogs you are going to need more punch. At least .25 cal but maybe .308 for a humane kill. A PCP air gun with 3000 psi scuba tank air source offers a lot of hitting power and some have a muzzle brake that reduces the crack. Run heavy projectiles to keep it well below 1000 ft/sec and it is quieter still. I have been looking at these recently as some in Idaho hunt deer and elk with PCP air guns in .457 or .50 cal. Inside 50 yards an air rifle is very effective.
Something like this might be suitable for groundhog day and not completely frighten your pup.
https://www.airgundepot.com/benjamin-armada-air-rifle-combo.html#BTAP25SX