So many things to work out and define there.
I’d agree though, flat fees for specific blocks of time to be “available” – ie. sober, near a computer and able to be awake, logged in and ready to go within 15 mins or so. Ability to veto or move particular ones if you have other plans.
Actual call-outs at minimum of double your usual hourly rate, with a minimum billed amount.
I personally can’t stand being on call so would quote something to really make it worth the disruption. The usual thing I’ve had in past jobs, of being a nominated contact for a week or more at a time completely screws up that time. Can’t go for a ride, can’t make any real plans, just stuck whiling away the hours at (or near) home in case the phone rings.