Not sure the details of your offer, but you have to justify your price. Do you know what the BPO came in at? Lenders are going to base what they accept on the BPO. If the BPO is too high, the price the bank gave you may be as low as they will go.
typically, a way out of their mess! I usually also offer UP TO 2K to help with moving expenses, and getting into an apartment or new place. If they want something in writing, I do a bill of sale for appliances or something like that. I used to make the mistake that I’d offer them $5, sometimes 10K! One deal we had to give the lady $15K or she wouldn’t deal, we talked her down to 10. Now, if I get a homeowner like that, I just say “I’m not your solution” or something along those lines, and move on. They usually end up calling back.