If she can throw equaly as well either way and is equaly comfortable doing so, I would let her throw right and bat both. If she throws left, she will be limited to 1st base or out field more and more as she gets older. Being able to hit from both sides of the plate is a definate advantage, with left being most desireable. If she pitches then left can be good, but not so much as to make up for the other potential limitations. JMO though, others may disagree.
Exactly! If you throw lefty, you essentially play 1B or OF (unless you pitch). There is essentially no advantage to throwing left handed. It's not baseball where you hold a runner. So the right hand tag is meaningless. And I've watched this game for a long time and can't yet recall any team making a 3-6-3 double play. So throwing lefty isn't that big and advantage from 1B either.