Your best bet would be to work back from what you know, or at least from what has actually been said about your William.

If you go to Guernsey and Belmont Counties, OH, where William 'spent his minority' you may find estate and land records of a Carn(e)s / Karns / Kerns who left a William as a son and at least one of the heirs. You might find a deed where William, living elsewhere, sold land inherited from his father in Guernsey or Belmont (such a deed might list other heirs and their residences, husbands of daughters, etc.).

Even if you found a Rev. War service record for a likely name, it would not tell you if that person was a relative of your William.

The story about Bunker Hill and captivity may or may not be strictly true. Family stories often have a grain of truth, but cannot be counted on to provide evidence.

Good hunting,
Jade