Based on your description one of 3 things are likely to happen.
1. Most desirable to you is that the first buyer believes the bank and goes away clearing the way for you to buy the home.
2. It's also possible the bank and the first buyer moderate their positions and come to terms, leaving you out. Many negotiations stay at a standoff for a while until someone blinks, and then acceptable grounds are found.
3. Finally the home could just end up in foreclosure. If this happens you may be able to buy it at auction for less than your offer price, but you'll need to be prepared to qualify to buy at an auction.
If I were the listing agent, in a situation as you describe, I would be pushing the first offer to come up to match your offer, the price the bank has approved, or to move out of the way. I believe itâ€™s generally in the sellerâ€™s best interest to complete the short sale for the highest price possible to limit any exposure my seller will have. Hopefully the listing agent feels the same way.
The bank usually does an appraisal, or BPO, when an offer is presented to determine the value that day in the market. I doubt that the bank will pay $350 to have a full appraisal done so unless the buyers want to I doubt one will be done. They might ask for another BPO to check the price with another Realtor and then make a decision. What I don't understand is if they are still haggling over the price I can't imagine they have a contract that is fully agreed to and so there is no need of a 2nd position offer (or backup as you call it). Why can't the bank accept your offer as the primary?