^I agree. If there is a tenant under a no pet policy in the other half of this duplex, then it's an absolutely hard no from me.
Let's suppose this is all true:
Mom has other options for her dog . . . don't fall for it. If she's in a rehab home, then the dog can stay at her place and she can get a pet sitter. Yes, it costs money, usually. If she's out of the home 4-6 months in rehab, then likely she can let someone house-sit/pet sit as a room in exchange for pet sitting arrangement. If Mom's recovering at home, then it doesn't make sense she doesn't want the dog's company. She can get a pet sitter, friend, or neighbor to come help feed the dog & take it out for walks.
Personally I think this is a made up story anyway. Color me cynical. Because I've had some major injuries and still managed to keep the dog in my home. I can't imagine telling someone to take my dog for 4-6 months because I want the dog around for company. My guess is that Mom is just fine and they either want to get a dog or they've already done it.
If you live in the other half of the duplex, then they likely only offered the pet deposit because they'll be caught for sure. Or they might be worried about the tenant on the other side ratting them out.