The remastered episodes cannot retroactively establish canon for the original series. As no registry numbers were visible for the ships in "The Ultimate Computer", there is no definitive answer to the question, only supposition and personal preference.
Yes, they very much can. Canon in an ongoing shared universe like Star Trek is fluid, with countless retcons and rewrites between "The Cage" and the next movie. For example, ST'09 establishes Kirk's dad as first officer of a starship, and according to Nimoy's Spock, he was Jim's inspiration for joining Starfleet. Although never mentioned or thought of in TOS, those new tidbits are part of canon and are "facts" in Trek's universe. Voyager pinned down the end of the TOS 5-year mission as being in 2270. What Sulu did "between scenes" of STVI was told in "Flashback". All 100% canon.
The '09 film isn't in the same continuity, even in the George Kirk sequences. The sequel will cement that contention, I'll wager, particularly if the rumors of dentated tribbles are to be believed, so referencing it is a non-starter. Later productions may add points of information but cannot alter what is shown. "Flashback" does that with dead/not dead Valtane. As such, its authority when speaking to TUC is suspect.
As for the remastered episodes, they are an alteration of an original work and not authoritative. At best, they are an interesting artistic exercise. If I repaint the Mona Lisa to give her a nose ring, is it now the definitive work? Suppose I've got the rights to release the the the old
Batman series on DVD but Nick at Night's gonna show them first. If I replace the Futura with a Tumbler, is that now canon? To cite a real world Trek case: What color is space in the Delta Triangle: Red or black? When "Time Trap" was first shown and then released on video tape, it was red. On the DVD, it's black. Which is canon? Black, because it's on DVD and represents the latest thinking on the show? Why not red? It was on the video tape. Which has primacy? The one originally presented or the one fucked up by someone who thought they knew better? To my mind, it's the original. TOS-R can no more establish
Lexington's registry number than it can alter the canonical look of the
Woden or the
Aurora or establish the
Antares was a modified cargo drone.