So, my brother-in-law happened by tonight on his bike. We went for a nice little ride just to stretch the legs out. The innertube handlebar tape worked really well. I didn't notice any rubbing off of the rubber onto my gloves yet. I'm taking the Peugeot out tomorrow morning for the commute. I'll keep a close eye on the situation before I commit to claiming any great success.
Maybe the hockey tape might later be in order. I don't have any of that laying around right now.