As Emil points out, $T_\Omega$ is not countably axiomatizable for trivial reasons: it has an uncountable language, and says non-trivial things about each symbol in that language.

That said, it is *equivalent to* a countably axiomatizable theory, in a precise sense: namely, you don't really get anything new after $T_\omega$. To see why, think about $T_{\omega+1}$: every formula in $L_\omega$ is also in $L_n$ for some finite $n$, so you've already added a constant symbol naming the relevant class; this means each constant symbol you add at stage $\omega+1$ is - provably in $T_{\omega+1}$ - equivalent to an already-added constant symbol. (Incidentally, this makes it easy to show that MK is *much* stronger than $T_\omega$, and hence much stronger "morally" than $T_\Omega$.)

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In fact, unless I'm missing something your hierarchy collapses right away - $T_2$ is essentially the same as $T_1$. This is because, in forming $T_2$, we can replace every "new" sentence with constant symbols $c_\varphi$ by an "old" sentence where "$-\in c_\varphi$" is replaced with "$\varphi(-)$".