Let me play devil's advocate, and none of us are assuming the that indeed the style in which the stage manager is communicating is NOT up to the par or the way or style the production manager would have a SM communicate. How funny we are all assuming this is the fault of a crazy Production Manager and not a sub-standard Stage Manager. There are lot of us on this board, we can't all be perfect stage managers.
Trust me, there are stage managers who do come across very poorly with their communication styles - and if the person who hired you is not happy with the style you are communicating, then take it as an opportunity to either improve yourself - or a great lesson to suffer the requests of fools gladly. If you want to work with this person, then you are just going to have to deal with their requirements. Don't like it, leave.
Why not just cc here on all correspondence? Is this some sort of power struggle between you and the PM? Why not just humor your boss. Ultimately, no matter what level you get at level, Stage Management is ALWAYS middle management and we are beholden to the people below us and held accountable by the people above us. If a producer, general manager, or whom ever wants us to jump through certain hoops as part of working with them - then sadly that is what we would have to do.
Now, the flip side, like everyone said, is to somehow confront your production manager about this request, which it seems like you already have done - and if she is going to make your life miserable if you upset her, then just don't upset her - so, cc her on the e-mails. As crummy as it may seem, it seems cut and dry.