Practically you can replace this particular ということで with として.
I cannot really explain the construction grammatically, but you can take this to be similar to (independent) participle construction in English (sorry I don't know the proper name for the construction; I mean something like Leaving the town, he headed for the north) as follows:
First,
で makes it adverbial:
So it is like embedding X as the description of the situation. If this 'theory' is correct, 君を「静」のキャラということで設定した translates literally as I set you, it being that you are the static character, or I designed you, your being the static character, which is I designed you as the static character in more normal English.
As you may be aware, there are a couple of related questions:
where:
- ゲストハウスの当番だということで : Being in charge of the guest house
- 私を暴発ということで殺す : kill me, it being that it was an accidental discharge
- 当面はノーサンキューということで(理解してください) : (Please understand) it being that I don't want it for now.