It's up to the bank, so no good way to answer that. Every bank works different and only they know what they are looking for
If I were the bank though, here is how I would rank potential offers
1. Cash offer - it's within 5% of the list price and not subject to any contingencies other than probably an inspection. This is a sure thing for the bank
2. Conventional financing - depends on the down payment and seller concessions. Large down payment with fewer concession means less hassle in getting financed
3. FHA financing - FHA loans take longer to process and the home needs to meet minimum property standards so sometimes FHA appraisals come back requiring repairs, which a bank is hesitant to do
As the saying goes, "Cash is king". Even if the other offers are higher, remember the home needs to appraise that high. And with appraisals these days a lot of appraisers are hesitant to bring in a higher value knowing that the home was listed at $256k. The price increase needs to be justified.
I sure hope you guys get the home, but if not keep your head up!