Any coach worth their salt, whether high school or travel, goes by the theory "If it ain't broke, don't fix it". However, if a kid "isn't getting it done" (batting in the .100s, numerous errors, era around 3.0), then for the sake of the team, a coach has every right to expect the kid to either fix it, or try suggested methods. In the case of a private instructor, for the sake of the kid's sanity, both the coach and instructor must be teaching the same fundamentals.
On the other hand, if the kid IS performing well, but the coach just wants to be a control freak, that's easy. Don't fight a losing battle. Just have your DD nod in acceptance of the coaches instruction at practice, and go through the motions. Then do what she knows works for her in games. As short as the high school season is, it's not going to mess up her mechanics. If it does, those mechanics weren't very ingrained in the first place.
Regardless, it's always smart to pay attention to any instruction - if only to compare and contrast. Who knows? You may pick up a little tid-bit of information that makes something work!