This is the major part of the problem. You ARE the parent, YOU are paying the price, YOU make the rules. Stop whining about the need for someone else be the bad guy by doing the PARENT'S JOB.
The equipment is there, you know it is there and affordable. TELL, not ask, her to use it or find another sport. Then you go tell the entire coaching staff of your decision, in front of other parents, and openly state that you intend to hold them liable if you discover they are not supporting and/or honoring your decision when your daughter plays. It isn't hard, just do it.
Since when is someone making a rule for the overall safety of players a bad guy? If I was in charge of ASA or PGF I probably would consider making it mandatory for certain age groups to start wearing facemasks.
I'm saying if there were a game rule in place it would remove the stigma of those wearing it and girls that probably should be wearing masks would wear them and reduce the risk of serious injury. Not everyone is as talented as those posters DD's who know how to pitch around big batters and field a ball hit back at them.
The game is changing. Bats are hotter. Players are stronger. 30-40 years ago I don't even think girls used helmets in softball. I'm sure a couple of players got hit in the head and then someone made helmets mandatory. I don't think its that crazy for that rule for facemasks to be put into place.