That's interesting concerning toxins... I wouldn't personally worry for skin testing because it is just a skin prick on the surface with tiny amounts of allergen mixture. But that doesn't eliminate the concern about What Is In There.
Eczema has many forms, many triggers. If the cause is environmental and from, say, the skin not providing an adequate barrier (something both dd2 and I have), then it can come from anything on contact. For environmental allergens, even if you get a positive test, the advice is: avoid and good luck. Allergens tested for are pollens, molds, etc. and usually not chemical tests (that I know of, I don't know what's out there, honestly). So, if it's chemical, you still have to figure it out and avoid it.
For foods, and especially at this age, even allergists who approve of "fishing" for allergies still might not test for much beyond the Big 8. Again, my limited experience might make me ignorant of what is out there, medically speaking. You might prefer the help of a naturopath, but frankly some of the "tests" and "treatments" they do for allergies leave me in a state of bewilderment..... but... some moms swear by them, but... yeah.... I'll drop that topic.
It sounds like the eczema can flare up and be extreme, so perhaps a physician can be helpful. That's not to say I would do an allergy test in your situation (and I am usually on board with allergy testing, in general). If there were a food allergy, I would imagine there would be other, obvious symptoms beyond eczema. But.... eczema in this form is not in my personal experience.
Causes of eczema can be beyond baffling, and many forms seem to have nothing to do with allergens. Still, if the eczema were frequent, severe and painful and I couldn't figure it out with my own research, you can bet that I'd at least investigate an allergy connection, and possibly with an allergy test.
How's that for a non-commital response?