Possibly. Unless there's something in a code of ethics for Constables, that gives cause for him to be removed if charged and found guilty of a crime. I don't know if even such a code exist. It's a good question for the Commonwealth Attorney.
"The county attorney shall give legal advice to the fiscal court or consolidated local government and the several county or consolidated local government officers in all matters concerning any county or consolidated local government business within their jurisdiction. He or she shall oppose all unjust or illegally presented claims."http://www.lrc.ky.gov/statutes/statute.aspx?id=24144