A majority of these ship's registries can be identified with known ships.
NCC-1831 USS Intrepid
NCC-1703 USS Hood
NCC-1672 USS Exeter
NCC-1664 USS Excalibur
NCC-1697 USS Essex
NCC-1701 USS Enterprise
NCC-1700 USS Constitution

The other three are unidentified.

An idea
In the late 2250's or early 2260's, the Miranda Class starship is commissioned. One of the first ships of this class is the USS Lantree NCC-1837. She served Starfleet for over a hundred years when she is destroyed by the USS Enterprise NCC-1701-D. At this point of her career, she is downrated to a Class 6 carrier.

I understand that the USS Carolina is identified as a Daedalus Class starship in the Omnipedia because of her presumed function-that of a freighter.
An interpretation of this vessel's career based on Mr. Okuda's facts. She is commissioned in the 2160's and serves to 2196 when her class is retired. Decades later, she is refitted to the position of a freighter and is serving Starfleet in that capacity in 2266. When viewed with the USS Lantree's career and the perceived facts of the Daedalus Class, this may make sense.
However, a major issue arises. Why would a freighter patrol the Neutral Zone? This is the stated position of this ship in "Friday's Child".

Based on the available evidence now at hand, I believe the USS Carolina could be a Miranda Class starship possibly with one of the registries mentioned above. (The registry of the USS Miranda is not known.) She is not identified as a Constitution Class starship in the Making of Star Trek.

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takeoffs are optional; landings are mandatory

Also, the USS Intrepid should be NCC-1631, not NCC-1831. It's possible that there are two seperate ships, or that the Encyclopedia was right all along, and USS Intrepid should have been NCC-1831.

And, are you sure that is the entire list? Because, I recall there being an NCC-1665, NCC-1667, NCC-1705, and NCC-1710 (which belongs to USS Kongo)

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"The things hollow--it goes on forever--and--oh my God!--it's full of stars!" -David Bowman's last transmission back to Earth, 2001: A Space Odyssey

The 359 Webpage

The other registries come from a time when dvds were not available and the quality of the screen capture varied.

Interesting how there are five numbers in the "traditional" Constitution range, a bit over 1700, four numbers just below that, and one number in the 1800s. The curiously small range of the numbers makes one think that there is something to the concept that these all are ships of the same class.

Then again, the small range of numbers might be because ships of roughly similar design age are likely to enter a period of refitting and upgrading at the same time. But the Enterprise is on that list as well - and she was only in for random repairs, not scheduled refitting.

Also, it's worth noting that a solid brass plate on the wall probably isn't the likeliest way to list ships currently docked. The list would have to be about something that ties together all those ships more or less permanently.

What WAS that brass plate about? Was there any header, any explanation of the contents? Does Frank have a screencap?

Since USS Miranda would be the most senior of the ships (at NCC-1833, probably, so as not to contradict the other fanfic regos), Starfleet would pull the same trick they did with the refitted Constitutions, i.e. call all the modernized subclasses by the common name Miranda.

Of course, one could counter-argue that the Constitution isn't necessarily the oldest ship in the Constitution class (since she isn't the lowest-numbered), so the Miranda need not be, either. But I feel it would be simplest to assume that USS Miranda, NX-1833, already existed during TOS...

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Dane