Can an experiment reach $\alpha \chi^{2/3}$ above one before radiation losses?

Reaching $\chi \sim \alpha^{-3/2} \sim 1600$ needs $\gamma E/E_S \sim 10^{3}$ ($E_S = m^2 c^3/(e \hbar) \sim 4.4 \times 10^{13}\,\mathrm{statvolt}/\mathrm{cm}$), but radiation reaction lowers gamma before the peak field is reached unless the interaction time is shorter than the radiation time. Determine whether any configuration, such as collisions of compressed $100\,\mathrm{GeV}$-class electron beams (Yakimenko et al. 2019), electron collisions with attosecond or tightly focused laser pulses, or aligned-crystal fields, yields a measurable number of events at $\alpha \chi^{2/3} \ge 1$ with an identified observable that distinguishes nonperturbative from perturbative predictions. An answer is a start-to-end design with event rates, or a proof that radiation losses forbid it.

Start-to-end simulations including radiation reaction showing a signal above background in a buildable configuration, followed by the measurement.