If $\alpha$ is too large, the electron and positron will annihilate before positronium can form.

As you note, this problem becomes non-perturbative, and I am not aware of any precise calculations. There is, however, a significant literature on the closely related "Z>137 Catastrophe" that happens if you have a large enough point charge. This is important in trying to understand if there is an end to the periodic table.

The hand-waving explanation is that as the electron and positron approach each other, at some distance it becomes energetically possible for an $e^+e^-$ pair to pop out of the vacuum between initial electron and positron. The $e^-$ from the pair will annihilate with the initial positron and the $e^+$ will annihilate with the initial electron producing photons. This will happen before any positronium state can form.

A theorist might be able to provide more details.

If $\alpha$ is too large, the electron and positron will annihilate before positronium can form.

As you note, this problem becomes non-perturbative, and I am not aware of any precise calculations. There is, however, a significant literature on the closely related "Z>137 Catastrophe" that happens if you have a large enough point charge. This is important in trying to understand if there is an end to the periodic table.

The hand-waving explanation is that as the electron and positron approach each other, at some distance it becomes energetically possible for an $e^+e^-$ pair to pop out of the vacuum between initial electron and positron. The $e^-$ from the pair will annihilate with the initial positron and the $e^+$ will annihilate with the initial electron producing photons. This will happen before any positronium state can form.

A theorist might be able to provide more details. There is a large literature on how electron-positron pairs are created in strong electric fields, and this paper on the breakdown of perturbative QED in strong electromagnetic fields is recommended in a comment.