Hontas Farmer is back
I have replied to your interesting and intriguing comment, love it, and it is pinned. The key to answering your question was knowing exactly what to look for. The theoretical work of deriving this result has actually been done already.
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.45.54 "Quantum electrodynamics with nonrelativistic sources. IV. Poynting vector, energy densities, and other quadratic operators of the electromagnetic field" by E. A. Power (Love that name in this context).
"For initial conditions given at t=0 it is shown that the fields are causal, i.e., for t<r/c the source-dependent fields are zero and the quadratic operators have only their zero-point contributions. For t>r/c they have both time-independent and time-dependent terms. The time-dependent terms, though transient, are shown to obey Poynting’s theorem. The steady-state part of the Poynting vector is related to the Einstein coefficients. The corresponding electric-energy density is related to the Casimir-Polder potential for a polarizable test body in the field of the source molecules."
Basically Veritasium and Alpha Phoenix in particualr demonstrated this quite nicely. That QED can give us Pyonting's theory has been long known. I doubt that Richard Feynman et al would've gotten a Nobel for it if it wasn't.
As for an experiment. I'm going to do some digging into literature about particle accelerator operations. Since I am certain they have to take account of every TINY effect to even carry out their experiments. In Veritasium's thought experiment causality enters the picture due to the sheer size of the thing, while in particle physics relativity enters due to the energies involved. The basic work on this has certainly been done.
The problem it is in documents like this that are so ...big.
https://www.jpier.org/PIER/pier28/07.9908012.Carron.pdf "Progress In Electromagnetics Research, PIER 28, 147–183, 2000"
Then there is this. Analysis of shielding charged particle beams by thin conductors
Robert Gluckstern and Bruno Zotter
Phys. Rev. ST Accel. Beams 4, 024402 – Published 14 February 2001
You know basically what we would need to look for are the types of things people who design particle accelerators worry about.