I think there’s a misunderstanding here, which perhaps has its roots in the idea that you can mimic a generic ‘British accent’. Most Brits would say the vowel in ‘path’ is either the one in ‘bat’ or that in ‘father’. It’s perhaps the most clear, or at least well-known, case of a north/south divide in accents, northerners plumping for the shorter ‘bat’ sound. I’ve even seen cases of dogs which have been trained with one version and do not respond (with their normal terrified whimpering) to the threat of a ‘bath’ in the other. (Also there’s my neighbour’s dog, who they trained with their South African accents, and would do nothing to a command of ‘sit’ from anybody English, but happily obeyed to an elongated ‘seet’.)