When 'batters' first became part of the terminology (if I recall correctly Michael Clarke might have been the one to first use it consistently), I cringed. But at that point I only really perceived it being a needlessly American addition to the sport. In recognition of the rapid rise of womens cricket though I have come to accept it and think of it in a positive sense in that it "normalises" that cricket is a sport for everyone to play and enjoy. Having said that I don't see why 'batsman' and 'batswoman' couldn't continue to be applied appropriately.
It's batsman or batswoman.
Batter goes on your fish unless you are Scottish, who prefer to batter Mars Bars instead.