Cat with eczema

Hello, our cat seems to eczema (or dandruff, I'm not sure if there's a
difference). Against my better judgement the cat's other parent
started bathing it, and it was a little while after this that the
eczema started. I'm something of a minimalist when it comes to cat
care, and throughout my childhood we never bathed any cats, and they
always kept themselves perfectly clean. So to be honest, I'm a bit
suspicious of the whole cat-bathing concept. The vet prescribed some
tea-tree-oil shampoo, so now the cat is being bathed with that, which
cures the problem for a short time, but it soon comes back, needing
more bathing.

I'm tempted to stop the bathing altogether, which will mean living
with a flakey cat for a while, but I wonder if the cat's coat and skin
will right itself and the problem will go away.

Any thoughts on any of this?

Incidentally, the problem seems to be a purely cosmetic one, and the
cat itself is entirely unconcerned (except it's not too fond of
baths).

Cheers!

Martin