You’ll find travelling in Aug is always going to be expensive hiring a car in Europe. Are you returning it in another country? If so, they’ll sting you even more for that.
With regard to your question on checking AWD, personally I have hired a car from Malpensa several times over the past 12 mo this through Avis and never been asked to prove an AWD code. I have used random ones too.
But, word of warning. Be careful when hiring as they will always look to add extras etc and before you know it your paying several hundred Euros for a basic rental. Also note there is a new pollution tax in the Milan area. You need to buy a pass once you enter an exclusive zone otherwise you will be fined down the track. Best part is the rental company probably won’t tell you, as when they catch up with you, they bang another 40euro fee on top of your fine.
Simple workaround, ask for a Hybrid as their exempt from the tax. If your not driving into Milan Centre, you won’t need to worry about the above.