I favor clear film straps; they're cheap, fast, easy, and effective. If you want elevation, use risers trimmed out of Gatorefoam or 1" x 2" stock and covered to match the background. The clear acrylic holders from Showcase and Superior are good, too, but more costly in time and treasure, and more conspicuous.
If the bat has collectible value for any reason, such as an autograph, then screwing it to the backer would permanently change its condition and reduce its value.
Please don't screw it unless it's a decorative-only prop for a sports bar or something like that.