The self tappers work ok in thin soft metal, but i think that they hold poorly in wood (I feel that the drilling function drills too large of a pilot hole in the wood for the screw to hold well) I would get regular pan head screws long enough to go through the band and half to three quarters of the way through the slats (you don't want to penetrate the inside of the barrel) Get two sizes of drill bit, one the same diameter as the screw threads and the other just a bit smaller than the diameter of the shaft through the center of the screw.
Use the larger bit to drill through the metal band. On this project you will probably dull several bits so buy a few, drilling at a slow speed will help to keep the bit sharp longer. A center punched dimple on the band will help to keep the bit from "wandering". A stop collar on the bit will keep you from drilling too deep.
Use the smaller bit to drill a pilot hole into the wood. Again, a stop collar will keep the bit from drilling too deep. You want your pilot to be almost as deep as the screw will penetrate.
Oak is tough wood. A little paraffin wax on the screw threads will help them cut in easily. If you are using a power driver, set the clutch so that the screw just comes tight to the band. You can always cinch it up by hand. That will minimize snapping the screws off as you drive them. If you can find USA made screws, you will likely have less breakage. (Try Van Wieren's Hardware. (I grew up in Holland))
Hope that this helps. Good Luck! It looks like a cool project.