A background check that didn't blatantly violate the Constitution as better enforced in most states would only be able to have a database of prohibited persons. Even that is questionable as you have no probable cause to assume that a private person selling to another private person who is not engaging in interstate commerce has broken any laws. So, even ignoring the utter lack of Federal jurisdiction, there is no cause to intervene here.
MA has "allowed persons" whereas NICS has only "prohibited persons" because databasing people who have broken no law in the rest of America is viewed as a violation of due process, invasion of privacy, etc...
So, no, UBC does not require a Federal Licenses, what it would require is opening up NICS to individuals which WILL be abused (NICS check to get a job, buy a car, get a credit card, etc...). NICS is simply not setup for any random shmo to use - in fact dealers are prohibited from using it for any other purpose of disseminating the information they receive from other than to complete a firearm transaction.
Don't surrender before the war even starts... We tried that already.
MA's scheme won't hold up in front of SCOTUS as is. I can't say exactly what they will allow, but even the 9th circuit wouldn't let what we have fly, much less some of the more conservative circuits or justices.