Extremely unlikely.

Cryptic would have major difficulty dealing with folks taking numbers from canon. Its not just a case of folks taking the Enterprise, its about ALL registries in Star Trek. I presume CBS wants to make sure that any future Star Trek games / movies / series etc. wont end up with problems because STO players are using registries within that range.

That said, I think it MIGHT be more interesting if Cryptic AWARDED unique registries within the four digit range.
It would have to be exceptionally rare so I'm thinking game events / tournaments where the top scorer(s) get issued with a single registry number they can use on any of their ship.