I would call their bluff, I had to.
A price was offered on my house way above the asking price, which blew other offers away.
Then they sort of kept finding things wrong with the property, and wanted to reduce by £15,000.00.
I just said ok, I will take the property off the market, do the repairs myself, then remarket it. And could I have an answer by close of business tonight please?
I am sorry this does not answer your question about equity loans. A bridging loan may be better, it costs more but is short term, if it is something you really want.
I do hope she finds a solution, it’s very stressful.