Can I cancel a bid from "private feedback" buyer?

I have had a "Private Feedback" Buyer open a case and receive a refund on an item I am sure (but cannot prove) was delivered. I want to restrict future Buyers who hide their feedback from bidding on my items. Their feedback is the only indication a Seller has of their past performance and how they treated other Sellers (via feedback left). I am aware that only a small percentage of Buyers do this but having been "ripped off" I want to minimise the chances of it happening again. Any advice would be welcome.

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