I look at it another way.
Lets say an uninsured pilot takes an aircraft for a solo flight - he is undoubtedly P1 and there is nothing stopping him from logging such a flight. Of course, the insurance company may not be happy, the CAA may pursue him for a breach of the ANO etc but there is no escaping the fact that he was the pilot in command and was P1.
Taking our intrepid pair of aviators, at any time they must have identified an individual who is P1; only one pilot can log the time. However, if they choose to be stupid and allow the uninsured to be P1 then he is still pilot in command but he has broken the rules by not ensuring that the aircraft is insured.