How do we deal with mom's doubly-false delusion related to an imminent out-of-state relocation from her home to assisted living?

You use "therapeutic fibs" to keep her in a calm mental state. You can tell her the AL is a hotel or temp apartment that she's stayinig in because some "unexpected" home repair is happening at brother's place (no water, not a/c, etc). Hopefully something like this will keep her manageable until she settles in. If she becomes a wandering risk (leaves to find you or your brother), then she may need to go into MC, which has tighter security to keep her safe.

Whenever she talks about her dog, don't correct her but instead redirect the conversation to something neutral and totally unrelated. Show her funny animal videos on YouTube or some News of the Weird story, etc. Just keep redirecting the conversation.

I learned a lot about dealing with my LOs with dementia from watching Teepa Snow videos on YouTube. She teaches techniques for more peaceful and productive interactions with people with dementia. I wish you alll the best.

Be in her world. You'll need to make up responses as you go along. You'll never convince her otherwise.

The good thing is that by the time this series of events actually happens, she may have forgotten about her dog and be on some (equally impossible) loop.

I'm sorry, I know it's stressful. My mom thought she was going to marry Elvis and watched out the window every day for him to arrive.

Your going to just have her think what she thinks. You will not going to be able to reason with her. She is in the past. May have to think up some little white lies when you get there. Maybe brother telling her he has company right now so he's putting her up in this nice place for now. Her dog? He is having a grand time with his dogs for now because this place does not allow them.