I’ve actually given this some thought myself!

If we can accept Galaxy Quest as a “names and details have been changed” tale based on real events that were experienced by the cast of Star Trek, then those events must have actually happened between the end of TOS/TAS and the release of TMP.

The franchise is essentially dead, but conventions keep it alive for the hard core fans. Nimoy, represented by Rickman’s character, is completely burnt of the franchise, having recently published I Am Not Spock. Also recently published were Franz Joseph’s blueprints, allowing the young fan in GQ the necessary information to lead the cast through the ship’s engineering decks. And the whole thing ends with an ad for a revival series - Star Trek Phase II, which would evolve into TMP. It even features the introduction of a new female crew member, secretly a Thermian (so I’m proposing that Persia Khambatta was in fact an alien).

So while a literal approach would mean you watch it before any Star Trek entry as a present day story, my preference would be to watch it between TAS and TMP, as a meta narrative of how the cast became triumphantly reunited before the production of the feature film series. Just pretend that all the characters are their closest Star Trek script analogues, and that the action takes place sometime in the late ‘70s.

I’d put it first since it takes place in the modern day. Unless you’re gonna do individual time travel episodes in their past time-frames, but I would keep those along with the rest of their respective series.