Update. Thanks for all the suggestions. They've been tossed into the pressure cooker at the back of my head to see how much juice comes out. This is my current thinking: I'll use a class name (which may just be Gryphon) rather than individual vessel name. However, the characters will give it a nickname.
Actually,
@SPCTRE one of my characters is loosely inspired by "Private Joker" from Full Metal Jacket, and I can see him naming it something like that just to annoy the CO.
Which leads me to yet another idea

. Each character calls it something different. When Joker got tasked with registering it he "accidentally" put in Shippy McShipface, which forces the CO to use the tag-code when communicating with other vessels/bases. Another character always uses the class name, a third just calls it "the ship" or "the mule", while yet another has some affectionate nickname.
Sailing round places like Portsmouth and Plymouth, I've see hundreds (thousands?) of boats, every single one of them (even little dinghys) with some personal name. And it often tells me more about the owner than the vessel itself - like a psych profile.
It's really interesting looking at all your different naming conventions, too. Thanks for sharing.