Getting rid of the dog could send her into depression.
It is her companion. Probably very well loved.
Unless SHE herself has said she wants to get rid of the dog or give it away.., it is no one else's choice to do so.
But yes, it is a burden upon all the dog sitters.
Could, she or you, "rotate" among different relatives, and care for her dog? So it is not only you. Or the neighbors.
Do you have other relatives or siblings there, who will ALSO do this duty?
I understand many people do not like her and she is a not nice woman.
But still, would you want someone to get rid of your pet, without your input, just because they don't like you?
Don't take it out on the dog.
The other issue is: you and the others, even her own son, does not like her.
The Dog is an 'innocent' bystander to all of this.
Or maybe, is there ANYONE, who would like to adopt her dog? IF that is what SHE wants?
has anyone discussed this with her?
What about any potted plants she has at home? Who takes care of that when she is hospitalized?
Or what about her home bills/garbage/home upkeep, when she is hospitalized? Who takes care of that?
At least, she is in an assisted living home. With a health aide. Otherwise, you all would need to pony up for that too.