Thanks for the drawing Skwinty.
Let's look at this. There are two terminals A and B separated only by superconductor, so V_a = V_b. There are two terminals C and D separated only by a superconductor, so V_c = V_d. Meanwhile: a and d are connected to a battery B2, via a resistor R2. Let's lump B2 and R2 together and call it B2'. b and c are connected to a battery B1, via a resistor R1. Let's lump B1 and R1 together and call them B1'.
B1' and B2' are connected in
parallel. By definition, then, V_b1' = V_b2'. If they're not equal "by design", a loop current will flow (ignoring the junction entirely) along the loop B1 - R1 - C - D - B2 - R2 - A - B - B1, and the voltage drops across R1 and R2 will
guarantee that V_b1' = V_b2'.
Which is so much as to say: ignore the details of the batteries/resistors/switches. What we have is a voltage V_b2' connected to AD, and exactly the same voltage V_b1' connected to BC.
With that out of the way, let's look at the semiconductor---the junction.
As has been pointed out many times,
if you're below a certain critical current then the junction
looks superconducting, so V_b = V_c. Therefore the batteries B1' and B2' are creating zero external voltage drop. The external work done by B1' is zero. The external work done by B2' is zero. There is no electric field across the junction. There is no voltage drop across the junction. No work is performed on an electron tunneling through the junction.
OK?
Now let's look at the other case. Let's suppose that the current is so high that the junction goes into "AC" mode. There
is a voltage drop across the junction. Let me call this drop K. Look at the diagram: that voltage drop is the same as the BC voltage difference, therefore it's the voltage supplied by the lumped-battery B1', so K=B-C = V_b1'. Look again: that voltage drop is also the same as the AD voltage difference. Therefore it's the voltage supplied by the lumped battery B2', so K = B-C = A-D = V_b2'.
So take an single electron that goes through the junction under this scenario. This electron has gone through
one or the other battery. An electron can get across the junction by following B - C - R1 - B1 - B
or by following B - C - D- B2 - R2 - A- B.
If it's the first loop,
then we see that the lumped-battery B1' has done work eK on the electron in getting it from C to B. Therefore the electron can only do eK work in getting from B back to C across the junction.
If it's the second loop,
then the lumped-battery B2' has done work eK on the electron in getting it from C to B. Therefore the electron can only do work eK in getting from B back to C across the junction.
If you don't like talking about "a single electron taking either path", then treat it as a current an use Kirchoff's Law. The current through BC (call it I_junction) is the sum of the currents through the B1' and B2' loops (I_b1' + I_b2'). The power delivered to the junction is the sum of the powers delivered by the two lumped-batteries (P_j = P_b2' + P_b2'). In the batteries P = IV so P_j = V_b2' * I_b2' + V_b1' + I_b1'. Both batteries have the same voltage K, so P_j = K*(I_b1' + I_b2') Multiply both sides by time and we go from power-current to work-charge, so W_j = K*(Q_1 + Q_2), and doing the same to I_junction = I_b1' + I_b2' we have Q_junction = Q_b1' + Q_b2'. So W_j = K*(Q_junction), and W_j/Q_junction = K, which is to say that the work
per unit charge is K.
K is the actual voltage across the junction. K is also the voltage supplied by whatever sort of battery you've installed. Your two-battery setup does not change this definition. The work per electron is eK, like we've been saying all along.
Go ahead and find an error in my math or my circuit analysis.
(I would point out that
if this were a series circuit, you'd get a factor of 2 from... but never mind, you already said this was the right circuit.)
(And don't tell me again that there is no voltage across a superconductor. I know that, that's why V_a = V_b. There is
sometimes no voltage across a Josephson junction---it depends on the current---but the case we're talking about is the high-current case where there
is a voltage drop across the junction.)