Correct! It's a Family Era piece and was carved between 1947 and 1962, probably in the 1950's and most probably carved by Horry Jameson, who started with Barling in the 1920's. The design is emblematic of Jameson's work. The TVF stamp stands for "The Very Finest", which Barling had registered in the 1920's but didn't much use until after the War. LLF is "long long flat", which refers to the shank shape and which is very rare, especially with a Quaint. The rustication on the panels is referred to as "chicken scratches" by Tad Gage, and it's an apt description.
There won't be a model number as each Quaint was unique. The carving looks very crisp, which is a plus. Too many of these have been subjected to buffers, also known as WPD's (Weapons of Pipe Destruction) so the carving gets badly softened up.
Check the underside of the stem, close to where it meets with the shank, to see if there's a stamping. If it says "Barling DESIGN" then the pipe is 1950 and later. If it says REG'd with a number underneath then it's 1947 to 1950. Not all stems were stamped, so there is no significance to the absence of a stem stamp.
This is a very nice find. The oxidation is pretty advanced, so you may find it better to have a new stem carved if you have to remove too much material to get down to virgin Vulcanite.
Congrats!