I emailed the FCC about the Batfish radar. I want to know if we need a license and what kind to operate it. I told them that the boat already has an amateur club call sign, that it is on dry land, and that the set is an antique (circa 1945) and may be one of a kind - I don't know of many SS radar sets in existence, let alone operational.
For a real ship, it would have to have a certain type of license (PG, DM, T1, or T2) and then someone on board would also have to pass a radar theory/operation exam, which would add a caveat/addendum to the license saying that the ship is licensed to repair and operate radar.
I will let you know what the FCC says.