I think your brother needs to call her on it when she gets manipulative. "Mom, your house is unsafe for you, let alone my child. I cannot risk her safety. I would be happy to support you getting professional help to deal with the problem and one day be able to have her in your home." If she can't babysit in his home, she can't babysit. If she laments that the grandkids don't visit, tell her that she needs to get help.
As for her siblings, I bet they know there's a problem, but you don't have to be her hotel. You can say, "Here is the number for the Ramada. I can't accommodate extra guests."
Hoarding is not fun, for the hoarder or the family. I really sympathize. My DD cannot stay overnight with my mother, either, though she can visit (some of the house is better than the rest, but the guest room is impossible).
You might need to have a sitdown with the aunts and uncles to explain that it's not about them. That your mom has an untreated hoarding problem and you feel like if you accommodate them for her, she has no motivation to change. It might feel like airing dirty laundry, but if the plain truth is just that - that you don't want to host her guests and it's not that you hate *them* then might as well say so. Maybe they can convince her to get the therapy and help she needs to get through the underlying problem.
Yes, your mom might cry, but I bet $100 you've cried about this, too. You may also want to seek counseling from someone who knows about hoarding, much like people go to Al-anon when their family member is in AA.
You may also weigh whether or not it would be beneficial to call some sort of elder care or senior services, much like people would call CPS if they knew a child lived there. Your father's health is at risk. Would she rather the authorities come in and take charge or have some say in the situation?