I didn't see a problem with the definite article before 'glimpses'. That said, I do like that your revision to 'any' raises the possibility that the N may have had no glimpses, which is nicely modest and definitely a move away from the definite. I guess extrapolating from that you could go more strongly in that direction with something like this:
for all my sitting, glimpses that it may
have caught of Nothing, doing what it will.
I guess another (rather functional) option for losing the definite article that makes it clear glimpse have occurred would be, "despite glimpses"
for all my sitting, despite glimpses it
has caught of Nothing, doing what it will.
If you're trying to lose the definite from the poem, though, I'd say that's another reason not to capitalise 'Nothing', which does seem to make it very definite.
I guess 'Nothing' is still bugging me here. There's nothing there, only there isn't nothing there, there's something call 'Nothing'. (Shopkeeper, "We've nothing in stock"; customer, "Great, I'll take six, please"). Lower case 'nothing doing what it will' gives you nice word play, which for me becomes less effective with the upper-case 'nothing': once nothing has become something, the fact that it can do things is much less of a surprise or paradox, which is what the word-play seems to hang on. Lower-case 'nothing' gives us actions but no actor. Upper case 'Nothing' seems to give us another actor. Rather than seeing through the illusion of a separate self, the N seems to have seen another self or higher power. Rather than no-thing, there's a thing. I guess maybe the point could be that 'Nothing' denotes 'everything', the interconnected universe (the flip-side of emptiness). And very possibly I'm being stubbornly closed to other readings
Anyway, I'd be interested to know why you prefer it or what, for you, it does for the poem, what you want it to communicate.