Disagreeing tactfully, politely, maturally...?

It doesn't sound like she will take your advice to heart, even if your suggestions are reasonable. When someone is so unapproachable, they usually don't value the suggestions of others. It sounds like she just gets angry when you try to talk to her politely, so she will probably get angry again when you try to give her advice on how to communicate. It's admirable that you want to help her, but I doubt she will listen.