Our bank uses our customer database to search for the OFAC and 314(a) lists. Therefore, I changed policy to prohibit the sale of any monetary instrument to non-customers. Branches are now balking because they have many non-customers coming in with large checks drawn on good customers (mostly lawyers) of the bank and the non-customers are wishing to cash a portion of the check and get a cashier's check for the remaining. Would the non-customer be considered the purchaser (therefore requiring us to check their names against the lists)? Or can we consider this an exchange and not have to check their names?

You could - if your bank is willing to take the risk.
Only you can decide that.
If one of those persons should be on OFAC you could be liable for a fine for issuing the check - the non-customer would be the purchaser.