If I were to keep the winsock to BNLS connected after completing the hashing for one account, could I perform hashing for another account without resending the Logon packets?

Yes -- the system is scalable, and each request sent to it is treated independelty. The cookie DWORDs present in many BNLS packets allow you to keep track of individual hashing requests on your end, because the server doesn't return them in any particular order. Have a look at the protocol specification.

http://yoni.valhallalegends.com/BNLSProtocolSpec.txt