You can't go through the hull because apparently the air spaces in the wood interferes with the signal. You could check BassPro/Cabelas to see what they offer in a clip on transponder mount that could be adapted for the Dory.

Do you have a dagger trunk?  Sailing mode or rowing?

If rowing could install the rudder mount thingys like the sailing version and make an adapter.

Can think of several ways to use the dagger trunk... 

I wouldn't want a permanent method on the bottom for trailering reasons.

Years back on a different small S&G boat forum I saw where the builder drilled a hole and filled it with epoxy/woodflour, like drill-fill-drill, but skipping the second drill. He mounted the transducer against the plug.

A wooden doubler plate inside the boat should take care of any structural issues. I'd make it at least double the size of the hole and bed it in place with epoxy/woodflour. The shape, circular or rectangular, would depend on where in the boat it was. Don't forget to round the corners, chamfer the edges, etc. As long as the transducer is mounted where it doesn't have to rest on a bunk or roller it should be fine for trailering with the doubler.

Laszlo

   Just purchased a portable fish finder from Garmin that has a large, suction cup mount. Much larger cup than other portable mounts I have seen before. Works well. The portable unit included a small battery and protective cover. Whole thing weighs @ 8 lbs. Hope to use it in my Wood Duck 14 hybrid soon. Transducers have issues going thru wood, so a mount is needed.

   Suction cup mount sounds perfect