Just because the BPO comes back lower does not mean the value of the home is lowered. An appraisal will be done by your lender to ascertain the value of the home as well. BPO are used by banks to determine a price point but it does not mean it is correct.
BPO's are not a very sophisticated mechanism for determining value, especially if the house is somewhat unique. The bank has no obligation to share the results of a BPO with a buyer. If it's lower than your offer I'm pretty sure you won't hear a thing.
If you're concerned about your offer, discuss those concerns with your agent.
Your offer should include an appraisal addendum that will protect you in the event the appraisal, not the BPO, is lower than your offer. Appraisals are required by lenders.I assume you are not paying cash?
Please feel free to contact me if I can be of further assistance: firstname.lastname@example.org or 240.529.0120.