This plastic version came in around the early '70s as I recall or maybe late '60s. It's the style I got when I first joined.
I expect with GDPR that's rather put the kibosh on making "registers" of "club" members public, even if they are now dead and always supposing someone has computerised the records
You could try writing direct to the RAC to see if they could tell you the date of issue at least.
Okay... got some info!!
This version is the metal and plastic one and was issued between 1960 - 1973. There is another version of it as well that had a plastic cover that went over the entire front, not just the center piece.
The number - 2355 - has nothing to do with the date it was made or who it was issued to; it supposedly means that the person the badge was issue to was given a 'road side help phone box key' - I guess not every one got them. Those with keys are earlier ones evidently.
The only way to know for sure about when it was made is through the paperwork that came with it. I have found one, like this one, that sold at auction, complet with its original box and paperwork, and it was dated 1961.
The one I got came with another badge, a Jaguar Driver's Club badge #1541, along with the grille bar. The guy who sold them to me said that they came off his grandfather's Jaguar XK120 that he bought in the early '60s. The car was totaled in a wreck back in the '80s - but they kept the badges, emblems and such.
I have contacted the RAC, hopefully they con confirm this info.