Honestly, the two shows have such different formulas in how the action is handled that I don't really see it working very well.

Like obviously, there are writers talented enough to make anything work really, but I think the two shows do have tones that would make an episode like this just be pure fan service.

I think it would work best as like either its own self-contained little mini-series, maybe one where the doctor finds himself in the Star Trek universe and winds up traveling and interacting with multiple eras. Or a one shot special that exists outside either shows season