This isn't all that uncommon, actually. It's possible that your dog will eventually stop having heat cycles without putting her through another surgery. I have a friend with a black lab that this happened to. She did not do another surgery and her dog had 2 heat cycles after the initial spay and never had another one. She doesn't have a uterus anymore, so there's no risk of pregnancy. I'm not clear on why your vet wants to open her up now, especially while she's going through a heat cycle. While a bitch is in heat, there is a lot more blood flow to that area and makes the procedure more difficult. Most vets don't like to do this, so is there a problem that your vet feels can't wait? I think my answer as to what I would do depends on that. If he doesn't think there's an infection or anything like that and just wants to do it so you have no further issues, I would say no and wait and see.
ETA: As to who should pay, I understand your feelings. I think the most you could do is get back the money you spent for the original spay and even that would depend on the vet. When you say dog home, are you referring to a shelter? If so, I think you're out of luck. I doubt a shelter would reimburse you or pay for another surgery, but I suppose there's no harm in asking.