would have to be a ship commissioned after SD 42527. Around that stardate, there was a Yorktown
with the registry of NCC-20045.
The earliest that TMP could take place was 2272. Memory Alpha has settled, largely, on 2273. We learn in a Voyager episode that the five-year mission under Kirk ended in 2270, and in TMP that Kirk hadn't logged time in space for 2-and-1/2 years.
And, there was a Miranda
-class starship with the registry of NCC-4186.
Finally, I don't know how you approach the Okudagram displays from "Conspiracy", but there are examples of ships with registries in the 8xxxx range, including the Ticonderoga
which has a registry of NCC-87270. (I notice that you are not including ships that were seen in the Okudagrams for this episode and "The Measure of a Man".)