I would also bet dollars to donuts that there is a even more refined version somewhere on the net.
You'd think so...but it's actually highly doubtful. Think about getting a screen grab from the TV and then enlarging it - all you'd get are just pixels. Any enhancements would end up just being an extrapolation. And as for getting the actual original source material from the producers of the episode itself - that almost never happens (and that website, Ex Astris Scientia, is actually pretty good at getting that stuff when available - it's probably the most comprehensive and dedicated Star Trek site on the whole Internet). Another thing to keep in mind is that it's just a background object - it's something someone (usually either the scene/set director, set designer or CGI artist) put in there either to add visual interest, keep the background from looking too "empty" or avoid it looking like a ghost town or simply because they thought it would be a neat thing to put in there. Because it's just floating in the background, and only on screen for about three seconds at the most, the CGI or physical model will be very simple and/or rough and when viewed in its entirety it'll be obviously unfinished at best and hard to tell what it even represents at worst. Because of this, simple quick photoshops are also very common, and I wouldn't be surprised if someone just real quickly photoshopped a real vessel (and once again, because it's just on the screen for a few seconds as part of a much larger scene, it's hard for the human eye and brain to register it as a cut-out moving in an extremely consistent, unnatural fashion).
Also, I just recalled the episode the Naval Patrol is mentioned in (one of my favorites from that particular series), and it's not made clear if it's an actual armed military naval force or if it's just the equivalent of a naval cadet corps ala USCG Eagle.