panic attack = anxiety attack - two names for the same thing.
It sounds like you may have generalized anxiety disorder, which is what I've been diagnosed with too.
With CFS/ME, it can be caused by a dysregulated central nervous system or a reaction to chemicals if you're sensitive to them (probably). I haven't really heard of wheat and diary causing anxiety before, but why not. I have problems with nitrites which are common additives in nuts and berries.
The usual treatment is an SSRI possibly with a benzodiazepine. Some people have problems with one or the other so you have to keep trying them until you get something that works. Zoloft is supposed to be one of the better tolerated SSRI's, and Klonopin one of the better tolerated benzodiazepine.
Since you already know you tolerate Klonopin, you can try taking the Klonopin 3 times a day - morning, afternoon, and bedtime, so you get constant coverage, and see how that works for you.
You can also go the natural route and try calming supplements such as magnesium, 5htp, tryptophan, theanine, etc.
Or you can do a combination of both - basically whatever works.
On the psychological side, you can also do meditation, CBT, or EFT (emotional freedom technique).
CBT is helpful for realizing the panicky thoughts that go through you head, that can make things worse. Plus sometimes, it's just nice to talk to a supportive person. Practicing meditation is good for learning how to calm your body quickly and learning how to live in the moment, and EFT is good for quickly zapping anxiety, past traumas and a zillion other things.
In general, any time you start a new medicine or supplement, start with a very tiny dose, much smaller than a normal dose, to test tolerance. Say like 1/8 of a normal dose. If you you're having problems, reduce the dose in half and half again, etc. until you can tolerate it. Then you can increase to the next larger dose and gradually build up to a normal or semi-normal dose.
Another trick in my arsenal is self-muscle testing. You can tell before you take something, if you can tolerate it, and also what dose to take. There are videos on YouTube that explain how to do EFT, self muscle testing, and I'm sure meditation too.
The mind-body-supplement stuff works for mild to moderate anxiety, so you may need some kind of med to control the heavy stuff and then use the other stuff for milder breakthrough anxiety.
Good luck - this should be a treatable problem, so don't give up.