I actually have the same knee problem as you, only that I have in both knees.
Ok it was only in the left knees actually. I got it for about 3 months. Unknown reason.
The same symptoms as yours: I can run, jump, squat to max just fine, but the (nearly) full flexion hurt.
I can sit in the ATG position or kneeling without pain when relaxed, but resisted extension from full flexion (slowly descend to full squat or slowly kneeling down, or slowly stand up from that) hurt. The pain seems to be in the middle of the joint (top of the tibia/shin bone), on my left knee it feels more to the left side, while right knee more in the middle. Not palpable.
When the knee was only in my left knee, it seems to be gone after some stretching, warming up by light knee extension at that hurt position, then slowly increasing stress and eventually I could do bw ATG squat with no pain. Usually I would do all that (stretch, warm up, bw ATG squat) a few time a day and the pain kind of disappear. It only come back after a long time sitting or no activity.
Here's how I got my right knee injured: because my left knee feel fine with enough warm up and I can squat ATG, I start doing ATG squat with barbell. One thing I didn't realize was that unconsciously I was avoiding the painful left knee and did three silly things: 1/Dive bomb into rock bottom position and bounce up from the calves (since no resisted extension at that position = avoid pain) 2/Push the knees out a too much(to make up for lack of hamstring flexibility) 3/More weight distribution on my right leg to compensate for my weak left.
The right knee pain kind of developed over a few days/weeks. And I didn't even realize that it is serious until my right leg's normal ROM is actually now less than the left (meaning more ROM that cause pain). Now whenever I full squat down I have to stop a little bellow parallel, put my arm to the ground for support and slowly release down (feels tight and hurt).
After a lot of thinking I suspect the cause is too much Varus (bow-leg, knees out) because I've always focus too much on that, and especially when squat bellow parallel I can clearly feel the torque on knees. So even if most people advocate pushing knees out during squat as if Varus is not evil, I think overdo it does cause problem.
I cannot afford doctor, so I did read a lot about knees problem, and yes these symptoms doesn't seems like it fit anywhere. The closest should be Meniscus tear (duck walk is painful), but it seems to be developed over time.
I hope you'll have a chance to have it diagnose better. I've had it for more than 3 months, and now both knees. It really sucks since I can do squat to parallel just fine, and the pain doesn't seems to get better or worse, but I'm still in doubt if I should squat or not. Please keep updating on the progress so it'll help people with the same problem.