I see two problems here. First, if you're walking a pet, you'll have to wait for an attack to be justified in shooting. Otherwise it could be made to look in court as if you were hunting.
Second, you'd have to be a superb handgun shot to handle the attack you described. Sure, it's only 20', but you'd have just a few seconds to draw and hit a fast-moving target. Add the risk of hitting the pet, and the stress level gets pretty high.
Under those conditions, nearly anything would do. I'd use a fixed-sight Ruger 22 semi-auto loaded with 40-grain hollowpoints.
1. They're accurate enough and powerful enough to kill a coyote at that range, although it may not do so cleanly or quickly. If you hit your own pet by accident, there's a chance it will live until you can get it to the vet, assuming you survive dealing with your family.
2. They're quiet.
3. The 22 LR shoots flat enough for the longer shots you mentioned if you decide to carry out offensive operations. Know your zero, as these guns tend to be zeroed at the factory for about 20m.
4. The fixed sights are fairly small and won't snag on a rapid draw.
5. Unlike your M-63, you can find Rugers in pawn shops for around $200, so if it ends up sitting in an evidence locker for a couple of years, you're not out much.
6. The ammo you'll need to practice enough to make that kind of shot regularly when the chips are down won't put you on the poor farm.
Just remember to police up your brass before you split.
Okie John