Hmmm, I've never heard or read of nor played a fat V-neck on a Les Paul. The different shapes are usually described either as a C or D, i.e. concerning their "roundness". Typical fat V-necks are found on old Telecasters, probably on Tokai copies as well.
As for the thickness it might help to look into the catalogues where Tokai sometimes uses the terms "'58 model" and "'59 model" to describe these Les Paul variants (but never "'60 model"). As far as I know, the higher priced models are usually the "'59" ones, at least in the early to mid '80s. For example my 1981 LS 120 has a much thinner neck than my 1984 LS 60 (and the other two LS 60s that I compared them to), so I would rather describe it as a '60 neck. But probably this doesn't mean that all LS 120 and above from that era have the same neck shape and thickness. :wink: So to half-way answer your question: the biggest necks on Tokais are probably the ones called "'58 models".
It's also important if the frets have binding and what finish the guitar has (nitro or poly lacquer), because this will give a different feel to your hands how smooth the neck is and how "fast" you can play on it. Fret size and height also play a part in this, e.g. when bending the strings or vibrating a note. This might also be different on the Les Paul Custom models where some were called "fretless wonders" back then, because their frets were very flat in comparison to the Standard versions.