.357 should be fine. If you have time to buy 00 shot shells for the shotgun, do that.
Your mom might find the dog without incident.
I knew a guy who lived up Sunshine Canyon. His small dog was taken by a cat off of his porch. He actually found the dog the next day, quite dead, but uneaten. The cat wasn't near it. The DOW ranger who was searching with him explained that cats sometimes/often/(who knows?) consider dogs as threats to their territory and will kill them for that reason, not necessarily to eat.
Good luck with that. Sorry about your mom's dog.