I'm not sure where to start with this problem, I suppose if the phase difference due to the circulating current is negligible then the total phase difference can be written

Where [itex]\delta[/itex] is the difference between the superconducting phase in the
two electrodes. Which seems like it must be relevant due to the similarity with the intended final result

Any help on how to go about this problem would be greatly appreciated

You must relate the phase to the vector potential A, where B = curl(A). This will include a factor of the flux quantum and 2 pi. Then you can do a line integral of A around the loop to get the total flux. Argue that the sum of the finite phase differences across the links must be zero mod 2 pi. You will find that the sum of the gauge-invariant phase differences is what you're looking for. A good reference is Tinkham's book on superconductivity, specifically chapter six section four. Your library should have it, or you can buy it for around $15 I think.