Short of the type of major surgery which most owners would find too daunting I don't think there is a solution to having a self-draining cockpit on the Privateer. The Shipmates, by the same designer, have a similar problem.
Of course one option, which you may wish to consider in the future, is to have a canvas cockpit cover made. This is fairly usual, amongst a wide range of types and designs of boat, and it should keep most of the rain out (ideally all of it) when the boat is unattended.
Alternatively you could limit the nuisance by installing a second drain hole a little higher up, comfortably above the loaded waterline. Some research (trial and error) would be needed to establish where the waterline is when there is water in the cockpit.
My own solution, in slightly different circumstances, is to just accept that the cockpit will not drain when the boat is afloat. But my boat lives on a trailer ashore when not in use, so I can then leave the drain hole open; and when in use I am aboard and thus I can bail out the cockpit as needed. But I am considering a cockpit cover for the future.
I initially used a loose bung, as you are doing, but then found a screw-in bung with permanent push-fit housing which was just the right diameter. Like this one, which looks similar and is from the same chandler, but if you are tempted to do likewise then do check the diameter of the hole first. If I remember correctly, the chandler has them in more than one diameter.
I do know that one owner of a Shipmate (Paul Brown) went to extremes, both to give himself more headroom and to create a self-draining cockpit. He cut horizontally right round the rubbing strake, and raised the entire deck by three inches, then made good the three-inch gap below the new deck. Then he raised the cockpit floor. But few owners would want to undertake that sort of a job.