In an ideal world this would be a good and reasonable step.... but I'd be very cautious and really really try all other methods before this one unless you are very confident that the other Mum is reasonable and will understand. I don't think she will do anything to disadvantage her own DD tho. And lets face it the child must get this behaviour from somewhere. And really what can she do except talk to her... she will probably do as you've observed say "oops" and carry on.
I say this Due to the danger of it exploding, eg the mum refusing to believe her darling is anything but perfect.. telling said darling.. who will resent it and being a bully will start taking it out on your DD and this could make the whole situation worse. Look at the theatre group situation and that is when she's on reasonably good terms with your dd. What would she be like when she gets her back up.
That's often why parents are advised to let the school deal with behaviour rather than approaching the other child's parents directly.
When I read that she never trashes her own lovely room, I thought that's a good sign that her trashing your DD's room and taking her stuff really is deliberate.
Tread gently... and keep your DD busy in activities outside of school without letting the friend know about it, so she has a chance to meet people without this baby Queen Bee getting in the way. Tell her to roll her eyes and say I don't know, my Mum keeps sending me to stuff but what can I do.
Make excuses not to have her round to your house and stop facilitating lifts and playdates. Blame it on work or something plausible.
If you can, break the cycle of your DD being dependent on this girl as her "bestie", because she is not this other girls bestie by any means. I agree with the PP who said she's treating your DD like a follower and not a friend. Encourage her to think of this girl as just one of the girls at my school. ie she's not the only one.
I think you need to keep a very close eye on this and do all you can to turn your DD's focus to new friends but without causing a backlash from the Friend.
Part of the problem is that having known each other so long, your DD is very familiar to this girl compared to other girls in her class and so she's more confident about acting this way with her.