What booster would you use?
Another booster up in the fairing? That would need to be quite heavy. It would be a totally custom thing for this specific mission. You would need a suitable storable propellant.
Leave the booster from the Ariane attached? The lh2 and lox would boil off after a few days.
What do you do if the deployment fails? SpaceX's dragon can't do space-walks on its own, it doesn't have an airlock. There would be no way to fix it short of developing a whole new space craft for that task.