I recently had a similar case, compensation for personal injury with recovery being made by CRG, and at the same time DWP realised they'd overpaid IB.
The CRG reduced the amount they were to recover by the amount of the overpayment. However, CRG were able to make the reduction before anything had been paid to them.
If CRG had recovered, and we'd lost the overpayment appeal, my plan was to try to persuade DWP not to recover the overpayment. I don't know if it would have worked, but I can't see how they could be entitled to recover both ways.
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