I feel like they're fighting the wrong fight. Rather than trying to make BS accept girls, lobby GS to grow. There's nothing wrong with the "boy" badges and GS should update rather than trying to force BS to allow girls.
I think the GS are trying to grow a bit or at least that is what I have heard from some of my friends about it. Hubby and I were talking about it and maybe it could be time for them to just change the names to something like Adventure Scouts and Pioneer Scouts or something that removes the boy/girl stereotypes if they want to open it to all.