I completely agree that giardia and food allergy are the two most likely culprits in your scenario. And whoever told you that it sometimes comes back as a false positive is exactly right. The way we diagnose giardiasis is by looking at a fecal sample under a microscope to find either the cysts or the trophozoites. The problem is that trophozoites usually stay attached to the intestinal walls, unless they are in the process of dying or reproducing. So if the existing trophozoites aren't dying or shedding cysts, or if they are only doing so in small enough numbers to have not been caught in the fecal sample, the test can appear to be negative when in actuality the animal being tested in infected. The good news is that the antibiotics/anthelmintic used to treat giardia is relatively safe with few side effects, so oftentimes if giardiasis is suspected, a vet will prescribe the medication just to be on the safe side.
It seems to me, however, that a food allergy is much more likely. By far, the most common food allergens in birds are corn, wheat, and soy, so it is probably a good idea to start by eliminating those COMPLETELY from his diet. You can't cheat and give him just a bit of this or that here and there. Any molecule can trigger an immune response. The other thing is that it takes THREE MONTHS for an allergen to be completely eliminated from the body, so you have to put him on this diet for a MINIMUM of three months in order to know for sure whether or not it's helping. If, after three months of religiously, fanatically keeping corn, wheat, and soy out of his diet, he's still itchy, you can move on to other possible allergens. Birds can also be allergic to food dyes, peanuts, or any number of other foods. He could possibly even be allergic to one or more of the seeds he eats. But start with the most obvious/likely foods and go from there.
The other thing worth noting is that animals with food allergies often respond really well to omega 3 supplementation. DHA and EPA help reduce inflammation, which in turn reduces itching, and helps normalize immune response as well. It might be worthwhile to buy pharmaceutical grade flaxseed oil (Vitamin Shoppe is one of the brands that passed consumerlab's inspection, and off the top of my head it's the only one I can remember) and give him 0.2 - 0.3 mL of that daily. It might take up to a month to see results, and you might not see any results from that at all, but it's worth a shot, IMO. Since omega 3s have so many other health benefits, I kind of feel like it would be a good thing to supplement regardless.
I also agree with the daily shower suggestions. Adding George's Aloe to the water could be helpful as well.
And I also agree with those who suggested getting a second opinion from another board certified vet.
So basically, this was just a really long-winded way of saying that I agree with what others have already said. Glad I could be redundant for you tonight.