Found a different loan agency that may be willing to do a refinance based on the value of the house after repairs are completed, rather than on the value right now.
It would mean no cash-out; all repairs would have to be completed by licensed and bonded contractors and would be paid out of an account. I'm quite okay with that, as long as I get to choose the work that's done (no way in hell I can afford a $300k mortgage) and it would be going toward fixing up the house.
Still a very preliminary option, but a hopeful one.