They are beautiful dogs and I truly hope you do not have to rehome them. Personally I think the demands that you listed are a little crazy, with the exception of money for vet care. Perhaps you and your mother could come to some sort of compromise. Like, instead of an electric dog door and new carpet (wth is that about?) you could simply crate train your dogs. I know a lot of people don't agree with crate training, but my opinion is that every dog should be crate trained because you never know when there will be an emergency situation where your dogs will have to be contained. Obviously, such a situation would be stressful for the dog. If they are crate trained, then the crate is their safe place and instead of increasing their stress level, the crate will greatly reduce it.
As for money for vet care, if you have income and not a lot of bad credit, perhaps you could apply for CareCredit. It's a credit card that's limited to medical, dental or veterinary usage. It's not real hard to get. When I got mine I was only working part-time and my credit was a few points below average.
I wish you and your beautiful doggies the very best.
