I am sure you will all laugh at the stupidity of this question but, here it goes. Why do you want want to avoid the bilge when running the transducer wire?
when I removed the screw the whole bled a brown water. I am assuming this is a result of the cleat screw not being properly seated with silicone and years of water seepage. My thought is to continue the hole, drilling completely through the gunwale so it drains into the bilge.
Sorry, my bad. You do need to remove the wet section if possible. Depending on the extent of the wetness there are products that "displace" the water and shore up the wood fiber. "Git Rot" comes to mind but there are others. At this point, photos of the area would help so you can get the best direction possible from the forum.Remember though, wood rot is like prostrate cancer. You can continue on for some time (often years) with both but left untreated they will ultimately consume the host.Good luck.