You can cross the airway of course subject to having received an airborne clearance, but of course you cannot fly on track within the airway as you almost certainly will not receive a pop up airway clearance for en route navigation.
Obviously given that you are making an air borne transit request to cross the airway at right angles there is no guarantee the clearance will be granted (although in my experience I have never known it to be denied) so you should be prepared to duck under the airway or accept whatever level you are given. Obviously depending where you intend to cross will give you a clue as to where other CAT users will be (in terms of level) and therefore whether there is likely to be any / much conflicting traffic.