I feel like at least one plot thread of that hypothetical film needs to involve Rutherford remembering that his family exists and then dealing with being different and having no memories.
Specifically, I imagine that with his implant, he's become everything his parents wanted him to be, and that horrifies him, as his parents almost seem happy their old son is dead.
I wonder how they'd rationalize it. I think if they'd done this, they should have done it like an anthology where each character has the same personality but is canonically a different person, almost as if Boimler, Mariner, Tendi, and Rutherford were character actors.
The maroon uniform era would be complicated; you'd need to figure out how to set it far enough so that they don't exist too close to either the TOS or TNG versions. I'd say the 2320s or 2330s ( roughly when Picard was in the Academy) so that they're all retiring by the start of DS9.
I'm not sure if DSC era proper would work well as a setting for a Trek comedy - I'm not sure there's enough to joke about for a whole season, and I think the overall tone of the 32nd century setting doesn't work well for a comedy. However, either a pre-burn future (mid 25th-30th) or maybe a few decades to a century after Discovery when the Federation's back on its feet might be nice.