What theory describes QED at $\alpha \chi^{2/3}$ much larger than one
In plain words
If the step-by-step expansion fails, a different way of calculating is needed. The question is what electrons and photons actually do in that regime and what experiments would see.
Precise statement
Assuming the loop expansion in a constant crossed field fails for $\alpha \chi^{2/3} >> 1$, construct a nonperturbative description, for example self-consistent Schwinger-Dyson equations for the dressed electron and photon propagators with a controlled truncation, an effective running coupling, or a semiclassical strong-coupling expansion. Give the resulting dressed dispersion relations and the rates of photon emission and pair creation, and show that the scheme reproduces the perturbative results for $\alpha \chi^{2/3} << 1$.
What would settle it
A nonperturbative framework whose truncation error is controlled, verified against known one- and two-loop results, and yielding testable rates.