Yanbu.
You are happy to make certain accommodations for this child but unfortunately the dynamic of your family - newborn, autism and a dog who is used to sleeping in DD's room don't lend itself well to too much change.
Obviously you can't have dog with newborn while dog is still getting used to another baby and needing to move a little further down the pecking order.
Autism isn't just something that is trotted out for fun and the disruption to your dd would cause massive consequences for everyone else.
The 'least worse' option is dog being shut in DD's room overnight and the girls sleeping downstairs, putting up with the whining from dog and hoping that dog doesn't wreck the room in distress.
You are doing the mum a favour and I understand that she might be worried about leaving her dd and having an operation, but compromise has to work both ways.
If she cannot compromise, then the only option is to say you cannot have the dd overnight.
And tbh, given you're having her regularly every week and so dog will be reasonably familiar with her, i might start being less available on those Fridays unless the girls really want to continue or she reciprocates with time spent at hers.
I am always happy to have my dc's friends over and help out. But only if and when it works for me.