batter

The verb comes from Middle English batteren, a frequentative form of batten meaning 'to beat' — built from bat plus the -er ending that signals repeated action. Separately, the noun for a beaten mixture comes from Middle English bature, borrowed from Old French bateure, 'the action of beating,' itself from batre, 'to beat.' A third sense, the baseball player who bats, comes simply from bat plus the ordinary agent-noun suffix -er.

The facts don't show a documented shift in meaning over time — the word seems to have carried its 'repeated beating' sense from Middle English onward.

It's a doublet of batture, since both descend from the same Old French bateure. It also shares its root with bat, the striking implement, through two different -er endings — one for repeated action, one for the doer of an action.