The first citation of 'off his own bat' in print comes from the pen of the celebrated cricket historian and statistician Henry Thomas Waghorn, in Cricket Scores, 1742:
"The bets on the Slendon man's head that he got 40 notches off his own bat were lost."
The 'Slendon man' was probably Richard Newland, the star of the Slindon Cricket Club and cricket's first great all-rounder.
It is worth noting that the phrase is found in print several times during the next century and all of the known citations are explicit cricket references - the other supposed derivations of 'bat' in this context owe everything to imagination and nothing to evidence.