While I'm obviously late to the party on this OP, I honestly think you're being a bit wet!
The piano may have been bought for your eldest daughter's use, but it's in YOUR house. Your daughter is 12, and not old enough to dictate what furniture is in YOUR house, and your Ex has no right to instruct anyone to do anything in YOUR house. So tell them both that the piano while it remains in YOUR house, will be used by anyone who lives in there, and while it does belong to your daughter, she must share it, in the same way that you would expect her to share other things. Maybe put the following question to her: 'If I were to buy an even better piano for your younger sibling, because you won't allow them to play on this one, how would you feel if you weren't allowed to play on it?'
To buy the younger children a lesser instrument seems unfair, so make a stand on this, it is not for your ex or your daughter to dictate what happens in this instance.
Maybe you should consider it in a different way? If for example your ex decided to buy your eldest daughter a dog of her own, would you be willing to do as he dictates when he tells her that she mustn't allow her siblings to play with the dog, or stroke it, or walk it, when the likelihood would be that YOU would be the one who would end up having to walk it, take it to the vets, and feed it. Of course you wouldn't! You'd say, if it's to live with us, then it's a family pet, and everyone will be allowed to make a fuss of it, and help take care of it.
Allowing either of them to dictate what goes on in the house that you also share with your new partner is ridiculous in my opinion, and is encouraging your eldest to behave like a spoilt brat.