Welcome to the forums, Hag. I'm a parent of a teen who has dyspraxia and dysgraphia, discovered when he was in early elementary. I'm not a professional, so take my advice only as that of a parent
What were the neuropsych's conclusions re the discrepancies in your ds' scores? Was it a private neuropsych, and was it completed before you sought help through the school? (I'm guessing from your OP that your ds is being evaluated for dyslexia through the school).
If the neuropsych report made recommendations for testing accommodations, you can request that the school honor those accommodations when administering the Cogat. If the school doesn't, you can use that to question the Cogat results. Is this a good or necessary thing to do? I would ask for the accommodations if the Cogat will be used for screening for the gifted program, but not if the purpose of the Cogat is to somehow help with the evaluation for dyslexia.
I'll also add that the Cogat is a learned ability test, not an innate ability test. Prior exposure to certain concepts (or lack of exposure) may result in the test missing a student who's gifted - my ds scored significantly lower on the Cogat than he had previously scored on the WISC, and as I read up on it (because I was beyond surprised lol!) I realized his experience isn't entirely unusual. When I asked him about the test right after he'd taken it, he was very excited to tell me about a few questions that he'd not ever heard of before but and used his logic to figure out (and it was very good logic lol!)... but his answers were off base simply because of lack of prior exposure to the concepts. I'd trust the WISC scores above Cogat. But again... I'm not a professional!
If your ds doesn't score high enough on the Cogat to qualify for the gifted program, and you feel it's a good fit for him, you can still advocate to get him placed in it. But I'd first want to focus on understanding the challenges he's facing - once you understand the root of the challenges and have a road map re how to remediate and accommodate, then you'll be in a much more solid place to both make a decision about the best program for him and be much better prepared to advocate to get him into the program.
While I agree with aeh re understanding the demands of the program you'll be placing your child in, I'd also add that for my 2e ds, being placed in a non-gifted classroom in elementary school was like listening to fingernails slowly scratching a chalkboard all day long for him - not only was he having to deal with his challenges, he was also bored to tears. The place he functions best is in a gifted classroom with accommodations. He needs accommodations no matter what type of classroom he's in, but he works and functions much better when the level of intellectual challenge is appropriate, and when he doesn't have to wait for students who need more time to understand concepts to catch on.
Re anxiety, for my ds, knowing that he needed help and not receiving it prior to his diagnosis took a huge toll on his self-esteem, and caused a huge deal of anxiety. Once we had a diagnosis, were working on remediation and accommodation, and once he saw that there were adults trying their best to help him, his anxiety disappeared. You mentioned your ds is dealing with depression and anxiety due to a recent move; I wouldn't discount that his hidden 2e may be at the root of at least some of his anxiety.
Good luck as you look for answers!
polarbear