"Me knowing that he was... " Is "me" the subject?
Solution 1:
The 'sentence' you asked about has no overall subject. It’s an example of artistic license where the rules of grammar get broken, not due to grammatical incompetence, but for some literary reason, typically resulting in fragments of a sentence being used, as your quote demonstrates.
The result here is that the sequence you asked about is not actually a full sentence, but a coordination of two non-finite clauses:
- Me knowing that he was up there.
- Them knowing I knew that if I busted in and dragged him out and bashed his head off, I'd not only be cashiered, I'd be clinked for life for having infringed the articles of alliance by invading foreign property without warrant or something.
Because there is no main clause, there is no overall grammatical subject.
Solution 2:
Note that I've expanded the quote to include some more context to make it clear that Faulkner is quoting someone relating an incident. Faulkner is reproducing the patterns of somewhat-convoluted speech. The main clause of the direct speech is
That was it
which is followed by two fragments punctuated like sentences that form an appositive to "it," that is naming the situation that "it" refers to. The two fragments are
[Their] Laughing at me
and
Me knowing ..., and them knowing ....
The antecedent to the understood [Their] and the explicitly-stated them is a roomful of French soldiers mentioned previously. The object of what they know is the clause
that if I busted ... dragged ... and bashed, [then] I would not only be cashiered ... [but also] clinked
which I've simplified by ellipsis. It takes Faulkner four prepositional phrases to describe what would get the speaker "clinked for life" (i.e., imprisoned for life).
Is it grammatical? Not strictly, but that often happens with the spoken word, which is what Faulkner has incorporated into his story. And it doesn't matter much if you've won a Nobel Prize for literature.