Maybe I'm missing something, but I don't understand why we can't agree on that point.
Following Jackson, the covariant definition reduces to the usual definition in a frame he denotes as K'. Later he defines K' as the frame in which the total momentum in the fields (using the usual definition) is zero. Since I have chosen a frame where E=0 everywhere, this is indeed that frame. Yes, if there were no moving charges, that would also be a specific
example of such a frame ... but in this case there is no such inertial frame where all the charges are at rest.
So again, I feel we are completely justified in using the usual energy and momentum densities for the electromagnetic fields. This caveat you brought up is interesting, but is ultimately unrelated to this problem as it does not require changing any of our calculations.