Rich,
I have one of the old Thompson Hiawatha forms. For what it is worth, I have been using 7/8" brass canoe tacks to attach the stringers that run along the bilge of these canoes. The planking is 3/16", the ribs are 5/16" and the stringer is roughly 3/16 to 5/16" thick, so you have about 3/4" of thickness when tacking from the outside planking through the stringer. I drill small pilot holes before pounding in the tack, which clinches when it hits the metal strip at the bottom of the slot in the form where these stringers sit when the canoe is being built. For the most part the tacks clinch OK, I use cherry, so it is a softer wood than oak. Dave's solution is also a viable one.