If I have found a buyer who is a close family friend but granted sole selling rights Wto the agent can I exchange unconditional contracts for sale 2 days after the contract ends after having the survey and paperwork done prior to this.
He found out about my house through another friend who is willing to vouch for it in a court of law

Your question is a bit confusing, but it sounds like you want to sell your house without paying the broker commission? If your contract includes terms for a commission even after the contract expires, we need to know what the exact wording is. Where this is might be important too.

Read the contract you have with the broker. It will spell out under what conditions you owe the broker a commission.