Quote by Jonathan Scott
"2] In this case with static fields and E=0, the covariant definition just reduces to the usual definition."
The case in Jackson where the covariant definition reduces to the usual definition (around equation 17.44 in my 2nd edition copy) is quite specifically the one where there is a frame in which all charges are at rest and B is zero, not the other way round. There may well be an analogous result for a static B field, but I don't know.

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.