Banks do not like to approve a lower offer than the first one they approved. However, they will look to their BPO/Appraisal - remember, they did one or more in order to approve your offer. If your appraisal is off from theirs, they may say no.... At which time I would hire a new lender and order a new appraisal rather than give up the house - assuming you do want to live there. Just tell the bank that's what you're doing and get then to agree in advance that if the second appraisal comes in low, too, that they will approve the sales price of the higher of the 2. That is so reasonable, that they will likely do it.
Appraisals are not scientific, and many are just plain wrong. Did you try to challenge the appraisal? Since you mentioned you have an agent, I am sure that the agent did that rather than risk losing the deal, right? I mean, if you agree with the appraisal, then it doesn't make sense to try to challenge it... but I am assuming that you and your agent think what you offered was reasonable based on the comps, otherwise you wouldn't have put in that offer, right?