Is the complexity class NP computably enumerable?
I can't make heads nor tails from what you attempt to do, but I know that the answer to your question is: no, NP is not enumerable.

First, let's fix what that means. I assume you are asking if

$\qquad\displaystyle \mathrm{TM}_{\mathsf{NP}} = \{ \langle M \rangle \mid L(M) \in \mathsf{NP} \}$

is recursively enumerable. It's not by the extended Rice theorem since the Turing machine $M$ that never halts is an NP-acceptor for the empty language, so $M \in \mathrm{TM}_{\mathsf{NP}}$, but there certainly are $M'$ with $M \subsetneq M'$ and $M' \notin \mathrm{TM}_{\mathsf{NP}}$.

Intuitively, no algorithm can separate $M$ (which has infinitely many equivalent brethren, by the way) from any other Turing machine in finite time, and thus can never enumerate them.