I don't know if "Not now" changes the depth of the POV, but I would argue that both of those last two sentences are repeating the sentence before it. If it's so close to his chance, we know he can't afford an injury right now. There's a few other instances of repetition I might eliminate that make for more concise narration.
so he could catch himself if he fainted or if his legs gave out from
He's going to fall with either of those events. You can use either of those alone to convey the full consequence.
it wouldn’t do at all if he fell and busted his face open
If he falls, is it expected he'll bust his face open? If he busts his face open, is it easily assumed that it's from a fall?
If we squeeze out some of those repetitions, we could get something like this that still maintains the deep POV.
He trailed his hand along the railing as he climbed the stairs so he
could catch himself if his legs gave out from under
him. He was too close to fighting for the title to ruin his chances by busting his face open.