Backdated ESA when it ‘overlaps’ with JSA

i have a seriously young client who came to see me after losing his ESA appeal. It transpires that after losing the appeal he claimed nothing for about 4 weeks and then claimed JSA. After about 6 weeks he saw a helpful G.P. who gave him a backdated med cert confirming a serious deterioration in his condition. He then put in a claim for ESA, backdated to the day after the appeal decision.
The DWP have paid for the four week gap betwwen the appeal and the JSA claim but then closed the ESA because he signed on.
It seems to me the DWP are having their cake and eating it. They won’t pay ESA because he was signing on but now also won’t pay JSA because he has provided a med cert!
What to do? Does he have to get yet another med cert and put in another claim for the post JSA period?
Thanks.

Is it possible to just withdraw a JSA claim that was made legitimately and was in payment? Would the fact that he now has a med cert for that period be a factor?