As Rob said, it isn't going to be watertight. Managing the water level in the cockpit is part of Laser racing. It could mean kicking water out of the cockpit. Or you may have to sail reaches with the bailer open while waiting for the start, then closing it right before the start.
That being said, could you find another bung to try? And maybe apply some sealant to where the center pin pokes through if it doens't fit tight enough.
Over what period of time are you getting 2-3"? When I'm sailing in lighter breeze and have the bailer closed I always have to kick some water out of the cockpit between races. Over the course of 45 minutes to an hour I might see at most 1/4" of water accumulate.
Yep. Too much, too fast. Have you tested it out of the water on a dolly? You can look under the boat to make sure the trap door is all the way closed when you push the bung in. If the trap door is not closed all the way then you have the bung pushed too far in on the rod and will need to back it off a littlew. Also, you'll be able so see how much water flows out if you put a hose in the cockpit.
The bung is tapered, (god that just sounds bad...) so it could be that it's not pushed far enough on the rod to create a seal. Either way you should be able to get enough of a seal with the device as equipped.