Re. rhymes with bath... I refer you to my [ahem] work of literature, below.
As well as Cath/Kath and path, I also thought of math (but you'd have to be American to get away with that, which may well be more unasseptable than being from sarf London), hath (if thou adoptest the Shakespearean styleee). There is also wrath, but I'd pronounce that 'roth', so probably doesn't count.
Incidentally, it's irrelevant whether you are from the north or the south, as long as you are consistent. So, if you say 'barth', you probably also say 'parth' and call yourself 'Carth', but if you are from God's own county then 'bath' will rhyme with 'path' and 'Cath'.
However, in considering this thread I have discovered that I have inconsistent pronunciation. I say 'parth' but 'bath'??? Am I a linguistic Frankenstein? Grew up in the South West (where pronunciation is somewhere between the two extremes), lived for a long time in the South East (where I somehow managed to adopt something vaguely approaching RP) and now live in GOC and find my vowels 'going a bit funny'?????