A fascinating question here... glad y ou asked!
I agree that any ship not seen on-screen can be interpreted in virtually any way.
However, I hate seeing too much repetition. I'd have been much happier had the TAS freighter not been used for Charlie's doomed rescue ship. (I WOULD have used that TAS freighter as the Woden, however!) And I'd have been happier had each of the three ships you mention been different.
If it were me, I'd have used this, exactly as seen, as Harry Mudd's ship. I'd have had something more along the lines of a "system patrol ship" (more along the lines of a small fighter, not intended for long-duration habitation) for "Operation: Annihilate!". And I'd have had something MUCH different for the Aurora.
Granted, I HATE the Aurora as seen in TOS... just totally cheesy. But the trick is to try to respect the original material as much as possible, which isn't what was done for TOS-R in this case.
The TOS-R "Aurora should have been larger, should have been shaped generally like the TOS Aurora... something that wouldn't immediately jump out and scream "not the same thing."
There are lots of ways to do that. (One of these days I may do a quick-and-dirty version of that as well just for "discussion" purposes... not as a renderable thing. But not tonight!)
But... in answer to your specific questions:
aesthetically, as a TOS spacecraft
Oh, it fits in perfectly well in this regard, except that I don't care for "blue bussards" and never will. I'm perfectly comfortable with there being many different "styles" of ship out there. I can see this as a popular "space minivan" design. I'd probably refine the shape and details a bit, but as a "first pass" I think that this is pretty good.
functionally, as far as its appearance as a spacecraft
Again, no problem, except that (a) it's not a "shuttlecraft," at least not as seen in "Way to Eden" and "Mudd's Women." It's a long-range vessel, intended for independent use, not operating from a base or mothership, and (b) it's a bit too sparsely-detailed to sell itself in a close-up view (but fine as a first-pass, low-detail model).
What if this ship had been used as the long-ranger shuttlecraft Copernicus in the TAS outing "The Slaver Weapon"?
No, I don't like that at all. It's actually quite a bit larger than the "slaver weapon" long-range scout, and it's not going to fit in any better than that does... plus it looks NOTHING like the ship seen on-screen (which is always a problem for me... I can accept "the thing we saw on-screen is an inaccurate representation of the 'real' design" but I can't accept "the thing you saw before is completely wrong in every way.")
Does the nature of this craft make it appear, in some vague respect, like a sort of TOS version of the DS9 Runabout? (Note: there is no indication that this remastered Class J ship is equipped with any military/defensive capability. It is never established that the craft has any transporter technology, or what its range would be, or how fast it could go.)
No, it doesn't strike me as a "runabout" at all. See, a "runabout" is more like a modern "Coast Guard Cutter." This ship is more like a "fishing boat" or a "pleasure yacht."