Don't see why not. The current map shows Banbury so CrossCountry cannot arbitrarily opt out of an 'Any Permitted' fare. If the route is permitted and they run trains on the route then they have to accept the ticket.
In fact I just tried a journey planner and it gives up plenty of options utilising CrossCountry between Oxford and Banbury.
Also a trawl through previous Routeing Guides show that Oxford - London via Chiltern Main Line to Marylebone has been a permitted route for a number of years.
The current map for Oxford - London Terminals is Map CS:
Of course, this map could be modified at a future date with a
Bicester routeing point added at the 'junction' (on the map) between Oxford and Banbury. That would then prevent journeys via Banbury as routeing rules forbid travelling through a routeing point twice on one journey.