I just skimmed through the thread and not sure if I absorbed everything so apologies for repetition.
In the past, I used to have these floating electrodes (I=J=0). The straightforward solution in my knowledge is given here but as pointed out is not applicable in dolfinx yet due to lack of real elements.
As a workaround, I managed to do fine by replacing the electrode boundary with a thin shell of very high conductivity. While I had a J=0 (i.e. natural boundary) condition, I presume it should not break things if one wants to specify a different Neumann condition.
No guarantees that it works but I think it is worth a try. If it helps, it will probably be the best workaround. The problem remains linear and the additional film of electrode has uniform potential which does not require a fine mesh.