This might not be a popular opinion either, but it really is not fair of your mom to rely on you all the time to pull her out of emergencies, any more than it is fair for you to rely on her. I mean, family is family. But, it sounds like your mom has wasted all of her good will already.
Still I am like you and can't imagine just sitting by to let the poor dog suffer. I don't know your whole story (although congratulations on the new baby-to-be) but it sounds like you are trying to keep your priorities straight in the best way possible.
What I would hope is that your mom can work something out directly with the vet. The vet will be able to impose much steeper consequences for her non-repayment (interest, credit dings, whatever) than you will, and it will be left to her to take care of it. I know nothing of the veterinary business, but I would hope that there is some sort of protocol in place to handle this type of situation. Some sort of payment plan deal where the dog can get her needs met and mom still be responsible for payment.
Anyway, I don't know you guys personally so take it with a grain of salt. It just seems like sometimes we end up feeling obligated to clean up other people's messes when there are better systems in place.
Poor dog. I've been thinking about her all night... Hopefully it is something simple and cheap and easy. Hopefully.