The division to the pre-UFP and UFP eras of this fictional universe is pretty blatant, though, and might be utilized here to advantage.
Yeah, Pike might have been training cadets aboard an old freighter. Many a navy today gives training on a mixture of old rust buckets, frontline combat vessels and dedicated schoolships, after all. And yeah, Leo Walsh and Harry Mudd might have flown one of those all by themselves, even if significantly larger crews were standard a century prior. Those old crews were more like inhabitants, after all, and increasing automation (remember the drone freighters of "Ultimate Computer" and "More Tribbles"!) plus faster engines (but Scotty still speaks of a warp 2 limit in "Friday's Child"...) would have made all the difference in operating practices.
But it's a bit simpler to assume that both Mudd and Pike had a somewhat more modern vessel at their disposal. Perhaps a more compact one as well, too.
A contradiction internal to TOS(-R) remains, though, as Pike's ship was apparently compartmentalized whereas Mudd's ship appeared small - both because of the crew count of just one or two, and because TOS-R adds visuals that match the small Space Kleinbus of the "Way to Eden" hippies. Thus, I'd vastly prefer Class J to be a set of parameters fitting a large number of designs, not a single specific ship design.