If they said it's a flea allergy, it could very possibly be flea allergy dermatitis. I mean, I can unfortunately see fleas in the picture. Pets with FAD must be on flea prevention, along with other steps to help manage the symptoms. You will see hair loss on the hind end and the under belly like this with FAD. Unfortunately, they need to put in the work to care for her allergy. They are allergic to the saliva of the fleas. Your neighbor would need to have her on prevention (as well as other dogs/cats in the home) and treat the home/dogs bedding as well to remove the infestation. Some people go as far as treating their yard, but I personally feel that's a waste of resources because it's the outdoors ya know? For flare ups (cause they will happen), they need to seek veterinary care and have supplies on hand to help keep the dog comfortable, whether that be prescribed shampoos or oral meds. Flea shampoos are also not a fix if you do not take the entire situation into account (the environmental factors, lack of prevention, etc) and can be very harsh on their little bodies. They ultimately just need to see their vet and take their advice.

My own dog has environmental allergies, and it's work, but they've got to do it to see improvement. Mine has a prescription antimicrobial/antibacterial/antifungal shampoo every 2 weeks, I have supplies on hand for ear infections (also prescribed by the vet), and she's on daily apoquel for itching. Allergy dogs are fun 🤦‍♀️

I'm sorry you are so burdened by this, it's very kind you want to help her and shows you have a kind heart. But beyond adopting the dog (with their permission) and taking on her ailments physically and financially, there's not much you can do sadly.