There could be several things at play here. The buyer with the FHA financing could be represented by a buyers agent of the listing agent or by the listing agent. When the listing agent told you that there were no other offers, that may have been true at the time. If he didn't let you know to come in with your highest and best, it's a good bet that the listing agent may represent the financed buyer. It's also equally true that he or she may not.
Here is the thing about REO listing agents: The majority of them are highly unorganized. It could have gotten lost in the shuffle. My advice going forward is to offer your highest and best contingent upon an appraisal to verify market value. This way you can offer higher without significant risk of paying too much and ask for an adjustment if the appraisal comes in lower.
Another approach is if this does fall out, in the financing terms of future offers, enter in a clause that reads something like "buyer to pay $1000 over any other bona fide offer. Seller to substantiate approved sales price upon acceptance." It works great especially if you're all cash. Insist the seller show evidence of other offers before going forward if possible and legal in your area.
Generally speaking, unless your offer is SIGNIFICANTLY lower than the financed offer, the bank is going to want to go with the offer with the least path of resistance, aka, an all cash offer. On a 203K though, the appraiser should take the condition into account that's why they are doing the 203K. At the end of the day, when financing, the seller doesn't care what the loan terms are as long as they get paid.
I hope that helps. If you have any other questions, please feel free to contact me.