Writing it on the fingerprint cards is not going to help when they run NICS. Even though the two checks are trying to do the same thing, they are completely separate ways of doing it. God Bless government redundancy, especially once you add in that your local dealer is going to do another NICS for the 4473. Anyway, you really are giving government drones way too much credit if you believe they are going to think about checking the fingerprint card for a UPIN when something comes up in NICS.
If I had a UPIN, I might actually write it into block 2a, along with my name and address. Even that would most likely go unnoticed by the government automatons, though, so the cover letter is probably the best idea. It stands out and grabs their attention.
P.S. If ATF or the NFA has a suggestion box, you might recommend that they add a place on the NFA forms for the UPIN, like there is on the 4473.