Do we have any recourse if family pit bull bites a person who is temporarily staying with us?

Probably several thousand dollars worth. You are fully liable for any injuries you dogs cause to anyone, no matter why they are there [except to commit a crime], but she is partially at fault for being an unreasonable person in trying to break up a dog fight, since a reasonable person would know that is dangerous. Her degree of negligence depends upon the exact facts of what happened, the type of dogs [determines amount of danger], what you were doing, whether she was a friend or just staying there a day or two, etc. It probably is best to go to an attorney who handles such cases, give them all the facts including what injuries are claimed and the medical bills and any lasting scars, and get an estimate of the dollar value of the case and then try to settle the matter on your own. Trying to make a quick settlement will be cheaper than dragging it out.

Dog bite cases are almost strict liability. The good news is that most lawyers take these cases hoping for homeowners coverage. With no coverage, the lawyer may drop the case, or at least take a nominal settlement. It could also be argued that the claimant was at least partially at fault. Negotiate a small settlement if possible.