Correcting some errors:

T is symmetric so the second term disappears

where z = [0 0 1]'

I can go no further and suspect I have made an error before this point.

I would really like to get an answer of the form:

but that seems unlikely as I can see no way to conjure up the partial derivative w.r.t. t.

Perhaps there is something special about the Maxwell Stress Tensor that allows me to do this?

This problem is related to my earlier post:

Calculus of Variations - Noether's Theorem for fields