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 $T_\Omega$.)
In fact, unless I'm missing something your hierarchy collapses right away - $T_1$ is essentially the same as $T_0$. This is because, in forming $T_1$, we can replace every "new" sentence with constant symbols $c_\varphi$ by an "old" sentence where "$-\in c_\varphi$" is replaced with "$\varphi(-)$".