Sacrificing as part of a command, to fulfill other costs?
No, you can't do that, because it is still essentially the same "can't sacrifice creature twice for 2 things", even if only one if an activated ability.

One way to reason it is to think about it with the stack. When an ability or spell is resolving from the stack, it has to fully resolve before doing anything else and it resolves separately from anything else resolving.

So, here you have Cruel Edict on the stack. If you let it start resolving, you can't activate the Aristocrat's ability until it finishes. If you activate the ability in response, the creature is sacrificed to put the ability on the stack. It resolves, then the Edict wants to resolve, and it needs its own sacrifice.