I agree. Your license is also a PPL and the rating is the same. You need to do the same PC. There is no need to inform the CAA. If you are truly flying for no reward you can let your class 1 lapse after 6 months as it then automatically becomes a class 2 for a further 6 months. After the 12 months you can either have another class 1 examination and repeat the cycle (6 months class 1, six months class 2, annual examination) or merely be examined for a class 2 which will have a 12 month validity. The class 2 examination may be cheaper but your class 1 will eventually lapse and be more difficult to regain.
However as I read it you are flying 'for' not 'with' an owner and that needs a class 1 unless he is a friend and no money changes hands except cost sharing. Post Sala I expect such arrangements may come under additional scrutiny. You cannot fly passengers single pilot over 60 (it is on the back of your certificate) although you can continue to do commercial ariel work single pilot and indeed 'exercise the privileges' of the class 1 when otherwise flying without reward.