Your approach is the correct one.
There are a few caveats that could explain the discrepancies in the results (one should, of course, obtain the same current flux between anode and cathode regardless of the method used):
the surface used to compute the flux should intersect with all field lines between anode and cathode, once, and only once. An example might be a plane that cuts through the head, with one electrode on each side. Another possibility is to create a box that encloses one electrode. In your example, with the sponge, I think you might have current entering from one side and leaving through the other and would actually have expected a total flux of zero.
the surface needs to be correctly discretized. For that, try to reduce the Minimum Edge Length property of the surface in the Explorer tree (after drag & drop), then Refresh Viewers, until the Total Flux value stops changing
I am not sure what happens if the surface is also a boundary condition for the J field (as in the case of your electrode). Maybe there are some discretization issues there.
In all cases, it often helps to visualize the flux of J on your chosen surface, by using the Surface Viewer
I hope this helps