batted and battered

The starting point is bat meaning ‘a stout piece of wood’ which surprisingly has an uncertain origin.  It might be a French borrowing, it might be Old English but borrowed from French, it might be Celtic, it might be coined from the sound the stick makes.  It is followed closely by batter to hit repeatedly which is a French borrowing.  The two are obviously linked.

So this could be part of the confusion which has given us to batter away instead of to bat away.  I battered away a tear from my eye.  Perhaps we think of the verb to bat as being just cricket.  Indeed the OED gives weight to the cricket sense, and minor mention to a couple of obsolete meanings to do with striking something, and an American dialect use of bat around to mean to roam around,  but neglects bat away entirely in its literal and figurative meanings.  To bat away a mosquito. To bat away an awkward question.

Or it could be the fact that batted and battered are very close aurally.

There is general evidence for this confusion which does not have much frequency in Australian English, for which we can be grateful. But the barriers may be crumbling.