I'm sure you meant to honor the crew and their ship (and the song), but I'm confident that Starfleet would not name a vessel after one of the most famous shipwrecks in history.
My thinking on the registries:
At first only Kirk's crew has done enough to earn keeping the same number with the letter. At that point, other ships like Excalibur, Lexington, Intrepid, and so on, are famous but not considered to be of status to keep their numbers.
The Star Trek Encylcopedia and some onscreen charts indicate another Excelsior with another number, NCC-27445. It has another captain by the TNG-era, but since Chakotay indicates that Captain Sulu helped him get into the Academy, I think we can assume that when NCC-2000 was retired, Hikaru Sulu and his crew got a new ship of the same class, but with some upgrades, but that new ship had a new number, since at that time, nobody had achieved anything like what Kirk had achieved. A lot of other ship names get re-used in this same way with new numbers for a long time.
I don't consider Star Trek made after 2009 to be canon for me, but I would consider Voyager to have done enough against enough adversity, and gained enough fame for her captain, that I could entertain the idea that NCC-74656 is a number that could continue to be re-used. Notably, Intrepid, the first ship of Voyager's class, has a number with no letter suffix, even though there have been several ships with the name Intrepid in Starfleet.
Regarding NCC-1305-E, I think we need to consider it an error for one simple reason: This would mean that, at some point, some crew of a ship called Yamato did something on the level of what Kirk achieved, and did it long enough ago that there have already been 6 ships with that name, when there have only been 5 ships named Enterprise. It might also indcate that the letter suffixes were in use before the number was given to If that had happened, then we should be seeing a show about that ship. So the regular number of NCC-71807 would make the most sense. (Although NCC-71806 would make sense, too, NCC-24383 must be another error, giving the registry of an older ship that was also called Yamato.)
