'Oh, I'm sorry, I can't quite hear properly, I'm sure there's supposed to be a 'please' on the end'....
'ah! There it is! Well done!' [hands over thing]
'Oh, dear. Never mind, I'll be here when you're feeling ready to speak to me'. [calmly moving away from the tantrum].
If she's used to getting her own way with her friends, it's tough being the smallest at home, as she's not the biggest. Doesn't mean she gets to manipulate everybody else through fear/noise/tantrums/entitled rages, though.
I've worked in some of the toughest schools. But I still used the 'I can't quite hear you' with six foot plus boys - and it worked. I was also calm and polite to them, my position being 'If you want something from me, ask me nicely, because I don't like being barked at any more than you do.' and 'I am always polite to you unless you are about to hurt yourself, others or damage expensive equipment. I expect the same courtesy back'. And I would tell them to come back when they were prepared to ask properly.
DP worked there briefly and said he really noticed how polite they were to me (and him) compared to their reactions to the barky staff. DP is also very polite to everybody.
The only person that I thought saying 'please' and 'thank you' must have caused physical pain to was my abusive ex. Maybe if his parents had done similar, rather than backing down, he wouldn't have grown up to be such a vile piece of work.
Don't punish your other children for her actions, but it's often possible to choose times when they won't be affected by her behaviour. They're the ones where being softly spoken but implacable have great effect.