I didn't get anything useful, but I uncovered a coincidence that's kind of cool: As I typed the most common plate number, 2F-3567, it occurred to me that the characters work as a hexadecimal color code, like you might use in Photoshop. (Geeky aside: As "normal" base-10 decimal numbering uses digits 0-9 and base-2 (binary) numbering uses digits 0-1, hexadecimal, or base-16, numbering uses digits 0-9 plus A-F. It allows values from 0 to 255 to be expressed as two-digit numbers: 2F in hex notation = 47 in decimal, E1=225, etc.)
The hex code 2F3567 translates to this color, useful to any artist trying to render "Death of a Mauve Bat."
I wonder if any of the plates were licensed to George Barris? I noticed some famous car plates were available recently including KIT-1 and ECTO-1A as well as a Batmobile plate. Only the Batmobile plate was sold out...everywhere...
Closer to "Gotham", 2F-3567 is a correct 1965 New York number; the 2F code would mean it was issued in Clinton County (the northeast corner of New York State).