I've watched this scene a few more times, and it's very good choreography, representative of the better work over the past few decades.
Notice how relaxed Philip is during the grappling—Royce Gracie had a similar looseness, usually used to supreme effect.
My guess is they brought in a stunt coordinator or choreographer or consultant with some military experience, familiar with Spetsnaz training and Russian wrestling, because that's Philip's background.
The fight sequence also has a narrative arc for Paige in particular, a hallmark of high level fight choreography, where it's part of her character development, and she gains a realization over the course of the fight, broken up into dramatic beats.
The way Philip anticipates Paige's movements, such as when she tries to kick back into his foot and he's already moving his leg back indicates the character has done this a million times.
- When Philip realizes Paige is biting his arm, he looks more annoyed than anything, and conveys with his face that he's just going to have to suffer it.
Releasing the grip would be arguably more unrealistic, especially in context of Philip's background.
Pressure of the rear naked choke would likely also diminish potential bite strength, and jamming the muscle even tighter into the mouth would be potentially advantageous. [This part just speculation, based on physiological assumptions.]