I'll be the first, but probably not the last, to predict that your return rate on these mailed questionaires would be very low. As a small businessman, I assure you I would not take the time to do it... My partner would, but she would just make stuff up.
You might try a test before you move to this method.
As for your regulator's opinion, as always, it depends on who walks through the door. If the EIC believes BSA is important and understands the current philosophy, he or she is going to say this technique is doesn't even reflect a good faith attempt to get the information you need to identify high risk customers.
Anyone who's ever manned a new accounts desk knows that the only day you are in charge of the relationship is the day the account is opened. Trying to get information later is a wasted effort.