I think this is pretty cut and dried.
I suggest, do some sums and work out how much it costs each month for exW to 'keep' DSD (include everything - percentage of mortgage/rent, bills, groceries, phone, internet, her clothes, her books, her mobile phone, holidays, school trips, dancing lessons, etc etc), then divide that in two (remember exW is responsible for half).
If the amount DH pays exW is less than that each month, he should pay half towards worthwhile extras if he can afford it. If he can't afford it and exW can't afford it, DSD can't have it, just like any child in a non-seperated family. DH should always be consulted first.
If the amount he pays is more than that, why should he pay for something he has already paid for once? The money has obviously been spent on something/someone other than DSD.
Simple as that.
Except there will always be exceptions. I, for example, would never ask exH for extra money. When things are cut and dried, they are simple and there is less potential for conflict, which is good for me and for DCs.
But DH's exW expects plenty of extra money, and gets it. Because, one: life's too short for the aggro and two: she'd still spend the money on herself and DSC would have to go without.