Joel, it seems to me that the sup of these metadefinable ordinals is exactly the first 1-stable. The key observation is that it's $\Sigma_1$ to say that a specific metadefinable ordinal is countable, and so there'll be an ordinal meta-defined by the ability to see this collapse map. This implies that the supremum $\sigma$ will have $L_\sigma$ ($\Sigma_1$-)pointwise definable. But then any $\Sigma_1$ formula true in $L$ with parameters from $L_\sigma$ is equivalent to a $\Sigma_1$ sentence. And so it must be true in some $L_\beta$ below $L_\sigma$ and hence in $L_\sigma$ by upward absoluteness.