Let's look at this a different way:
You buy it and it is registered as model # "320C-9-BSS-MS-MA"
Three years later you sell it to a dealer and have thrown the box away.
Said dealer records it in his BB as Model # "320C" as that is what is marked on the gun. He sells it to John Smith and records it in the MIRCS system as Model # "320C".
A few years later someone does a trace by S/N and they find TWO guns with the same S/N, one as "320C-9-BSS-MS-MA" and one as "320C". They don't know that it is the same gun, so they come knocking at your door.
Keep in mind that it is perfectly legal for a mfr to re-use S/Ns for different model guns, so the S/N and Model # are a set that should be traceable.