I don't see the difference between no limit and refinance the house so I voted refi.
I would do WHATEVER it took to get the correct medical help for any one of my babies. Right before my Rocky, a non grey passed in '04, I spent 8K in one week only to have him die.
Whatever it costs... it costs. And no, I am not in a position too, who is? I would just deal with the financial cost as it happened. That's what 0% credit cards are for.