Well, I AM A Trekkie (hardcore TOS purist), so if you'd like I'll be happy to help you provide your watchers with a TOS ship download that at least adheres to TOS's canon.
TOS never actually contradicted itself outright, and throughout the run of the 79 episodes, they DID establish nine out of twelve of the names that ARE canon. We only know two of the registries as canon (1701 and 1017), but following another canon source (Matt Jefferies, the man who designed the Enterprise) by using his registry number system, the rest of the ships would have registries that count upward from 1701 onwards. Jefferies' system was that the 17 referred to the fact that the TOS Enterprise was a member of the 17th starship design made by Starfleet, and the "01" referred to the fact that it was the first of it's class. (This meant that the TOS Enterprise was actually of the "Enterprise-class," which was followed even until they made TWOK).
Here's the list of CANON names of "Connies" from TOS (in alphabetical order):
Constellation - 1017 (it is generally assumed that, like the Enterprise-D, the Constellation was named after an older ship, and that it also bore the older ship's registry)
Enterprise - 1701
Yorktown (not in the episodes, but one that Roddenberry established)
Although giving the other ships numbers would be conjecture, as long as it follows the basic count-up (1702, 1703, 1704, etc) I think you'd be fine.