If there isn’t enough difference in them from a design standpoint, how about artificial limitations? There’s little point in creating three crews unless three ships can do things (and tell stories) that a single ship could not.
How much mileage is there to be got out of the idea of one particular ship’s computer being outfitted for maximum computer capacity? They are exploring an entirely new quadrant, after all; wouldn’t they need a vessel with as much ability to store new information as it could hold? They will be encountering new cultures, new languages, new diseases, new stellar phenomenon, new planets full of species, each culture with its own incompatible computer interfaces, so tons of new technology to analyze… perhaps one vessel would be fitted much more for advanced researchy type stuff and analysis rather than the same-old-same-old Alpha quadrant stuff found in the old memory core. Such a design philosophy could even begin to affect the character of the crew that manned that ship.
If we’re speaking of Trek triumvirates, why not emulate the best of all, the Kirk-Spock-McCoy triumvirate, in the three-ship delegation? A ship with an enhanced computer core could be the “Spock Ship” in that it is principally concerned with the future and contains the freshest and most exhaustive data they have yet found. Its limitation then becomes obvious: without its standard computer core it lacks the data to fully treat its mixed-species crew’s medical problems or recreational needs, and while armed, its information is valuable so it is generally not inclined to stand toe-to-toe for a prolonged pounding.
Thinking along those lines, the “McCoy Ship” would have all of the old computer core records, for the sharing of Alpha Quadrant culture with any new allies it meets. It would have the best facilities for food, medical, and creature comforts. It has more replicator patterns for useful and practical real-world stuff than the other ships. Such a ship would essentially be an island of Alpha Quadrant and might be more conservative, compassionate, and human-centric (as McCoy was).
The “Kirk Ship” would be the warrior vessel and the flagship, with plenty of extra computer space, containing backups of some of the “Spock Ship’s” data, and more capable of defending itself. It would be the ship that stayed behind to be captured by aliens and made to fight in a tournament, and would end up sleeping with that green girl with the go-go boots. Somehow, once per episode, the “Kirk Ship” would get its shirt ripped off. Okay, okay, I kid … but it would be the ship that holds the line if the other two need to run.
Perhaps such a division of character among the three ships would be easier to imagine first, then choose what models each should be.