Anyway, IVA Richards is clearly number one.
Between Weekes and Barrington, I think most probably Weekes doesn't have the career to be better than Barrington, he was unquestionably rated higher and it wasn't even close, man was deemed a monster by both Sobers and Bradman but he was healthy for only one of his three away assignments, a thigh and hand injury in Oceania in 1951, that was promptly exploited by Ray Lindwall and Keith Miller with a ruthless bouncer barrage that would make Roy Gilchrist blush. Of Course, the Sinusitis tour to England in 1957, where he was under the knive for five times in a few months span, and that is not a good thing, throughout the tour it resulted in constant migraine, constant pain, everlasting exhausion and double/blurred vision, none of that is ideal when facing the attack of Fred Trueman, Brian Statham, Trevor Bailey, Jim Laker and Tony Lock. Though, he did play some great knocks this tour, there was the flashes of brilliance, he made a 90 against England that Denis Compton termed "an inning of pure genius", a 64 against the almighty Surrey team of the 1950s when Loader, Bedser and Laker were bowling and another hundred against Trueman, Tyson, Wardle and Bailey. But Obviously, he was too far from best physical condition to keep it up consistently.
Regardless, I don't count these serieses against him, infact the 90 and 105 in 1957 in such poor health I regard as quite good, but sadly that means Weekes only has one tour of England when healthy, where he averaged 56 in 4 matches, 3 half centuries and 1 century, averaged 80 in the tour overall including First Class. But that's it for his away work. Nothing in Australia sadly. He also retired at 32-33 so ofcourse the injury problem was permanent, Barrington just seems like a more proven player and more importantly, more reliable and less injury prone, and if that matters for bowlers, it should also matter for batsmen.
