The only thing I can recall "on screen" that supports Franz Joseph's NCC numbers is some of the communications chatter from the Epsilon Nine station in the opening scene of ST:TMP - They specifically mention the "scouts" Columbia and Revere (I think it was Revere, I'm at work and can't easily check atm), plus the dreadnought Entente - all of which are referred to by NCC numbers which match the Technical Manual.
I'm not aware of any dreadnought
Entente being mentioned, here's the com chatter:
LIEUTENANT: This is comm station Epsilon Nine, calling U.S.S. Columbia. Come in Columbia. Respond!
COLUMBIA VOICE: (too faint to understand)
LIEUTENANT: This is Epsilon Nine. Am boosting output. How do you read this?
COLUMBIA VOICE: All right. (too faint to understand)
LIEUTENANT: Scout Columbia NCC six two one to rendezvous with Scout Revere NCC five nine five on stardate seven four one one point four. Further orders to be relayed at that time. Signed, Commodore Probert, Starfleet. End of transmission.
COLUMBIA VOICE: All right. (too faint to understand)
NCC-621 is a scout ship and tells me it should be an Oberth Class vessel.
Apropos stardates: By the time of TMP it always seemed to me they had somehow decided to settle with "1.000 stardate digits = 1 solar year".
Considering it's a five year mission that probably started at 1277.1 it would have ended around 6.277.1 (i.e. excluding TAS).
Bob