Exactly what did they diagnose you with? "No depth perception" is pretty broad. When I was applying to the AF, and throughout my career, I always failed that DP test that uses the little circles in a row. I would always have to go see the eye doc and do some special tests to prove that I really did have depth perception, even if it's not perfect. Visual acuity is measured very precisely using the 20/xx scale. However, there is no gradation for measuring DP, only pass/fail tests.
Eventually, someone decided that I ought to be evaluated, and was diagnosed with a microexotropia with monofixation syndrome.
I was immediately grounded pending further review at Brooks. Under the AF regs, all depth perception waivers must be evaluated at Brooks in person. Only Brooks can issue waivers for DP deficiencies. After 2 days of testing, I was granted a waiver and given a pair of glasses to help correct the monofixation syndrome. They really can't fix the microexotropia.
But bear in mind that by this point I had 14 years of service, and had been an instructor in 3 different AF airframes. I'm sure that was taken into account in their decision to issue me a waiver, especially for something that was fixable with a simple pair of glasses. Admittedly, I never wore the glasses for takeoffs, landings, etc. The only time I needed them was during air refueling.
What tests did you fail? Presumably the "circles in a row." But did they then give you the polarized glasses and "grab the fly's wings" test? Or use the glasses with the pictures of animals and ask which was closer (actually a test for kids, but they use it for adults, too)? If they only did the circles, then you might be able to ask for further testing using alternate tests.
Good luck, PM me if you want.