First welcome to the site Drew.
Nope, it's perfectly legal to use it in any league or tournament. The only thing USBC approves are balls, and whether or not a clean can be used during or before or after competition or at all on a ball.
As for the athletic
tape, I recommend against it, speaking from personal experience. The glue bleeds through and can get in the finger holes and can cause all sorts of issues. If you want a
tape to wrap around the fingers Vise's Bio-Skin is probably the best bet. It stretches and allows you to still bend the fingers.