This is definitely a 500 Model J from early post-war production. Even without the numbers, it is easy to tell a 350 from a 500. The 500 has numerous visible differences to the frame, engine and cycle parts compared to a 350. Shape of the cylinder head, size of cylinder, position of the front engine mount, centre tube bottom fixing, rear frame tubes, rear brake, rear stand are all easily distinguishable from photos. Not so easy to spot are different gearbox case, engine plates, air filter box, chainguard, regulator mount, pillion footrests and other smaller parts.
One published Enfield history book specifically states that the 350 and 500 had identical cycle parts, which is simply not true.
If I wasn't so far away from it, I'd be bidding.