It might be a dietary allergy.  Try feeding her on only chicken and rice for
four weeks and see what happens to her skin.  Yes, it's hard, she won't want
it every day, but see what happens if you do it.
Don't give in to her if she says she doesn't want it,.just let her go hungry
until she eats it and see if she improves.  There is  nothing to be allergic
to in chicken or rice.

Skin problems are very, very difficult to sort out. but can usually be done
in the end.  Take hope.
It can be dietary, as I said, but can also be do with chemicals used in the
house.  Carpet fresheners, and all sorts of silly things we use to make our
houses clean and smell nice.  Sad to think it often harms our animals.

Tweed

Is there an elevated eosinophil count?  You get a proportion of
eosinophils to other white cells on the CBC, but a quantitative
eosinophil count is a separate and not too exotic test.

An elevated eosinophil count is a nonspecific indication of some
inflammatory process. In humans, one would rule out obscure infections
first, but then the suspicion might go to a cutaneous vasculitis. There
is a human disease called Churg-Strauss syndrome, which is characterized
by the triangle of eosinophilia, eczema, and asthma, sometimes with
kidney involvement. Churg-Strauss isn't curable, but usually can be
managed with corticosteroids and cyclophosphamide plus (probably not
practical for a cat) plasmapheresis.

I'd also wonder about the results of some general autoimmune screening
tests, such as immunoglobulin electrophoresis.  The academic specialists
here might just as likely be rheumatologists as dermatologists, as
rheumatologists are the experts on figuring out WHY an inflammatory
process is going on.

If it hasn't been done already, a skin biopsy might be very informative,
if you can find a dermatopathologist to interpret the slides.
Dermatopathology is a subspecialty even with humans; it's probably wise
to ask the dermatopathologist how the biopsy should be taken and how the
samples should be preserved BEFORE taking a sample. This sort of skin
biopsy is a minor office procedure for humans.

The drugs mentioned below are just a few of those that are used in
combination with, or as alternatives to, corticosteroids for many
autoimmune diseases. In humans, a low dose of an immunosuppressant like
methotrexate allows you to lower the corticosteroid dose.

Now she's one very greasy kitty!