@ Crazy Matt
I think the tombstone engraving in WNM remains a subject of debate. One camp (to which I belong) believes "1277.1" to refer to the stardate James T. Kirk had been promoted to the rank of captain ("C"). I believe this theory holds water because the time span engraved here ends in what is the current stardate of the episode. Essentially it may be a sick joke of Gary Mitchell to emphasize on the short career of James "Rice" (academy nickname) Kirk as a starship captain.
The other camp believes it to be a birthdate (old Federation calendar? not explicitly listed as "stardate"!), much like the birthdates in Mitchell's and Dehner's file we see earlier in this episode onscreen. The thing is that the different digits in those files correspond to the difference in actual age of Mitchell and Dehner.
But there's no way, IMHO, to align "1277.1" within such a hypothetical system. If it were not a different time measuring system, one would need to ask why they didn't use stardates in "The Cage" because obviously Mitchell and Dehner had been born prior to events in this episode.
I'm pretty certain there is no stardate mentioned in "my"
file on Talos IV.
Is it possible you are thinking of the Mitchell and Dehner files from WNM instead?