Boy oh boy, are we making a meal of this.
If your real question is 'does it matter that the FPL and techlog say different things', then that's the question you should ask. And in answer, I would say that the Pilot in Command field on the CA48 flight plan form is not part of the transmitted flight plan, so I suspect has very little legal standing.
If your question is who is P1, then the answer is whoever you agree is P1. And if you don't agree then you are asking for trouble, because the only time it will matter is when something goes wrong, and at that point your unwillingness to agree up front will become part of the problem.
If you're unwilling to confront the issue, then you can simply ask the client if they intend to log the flight. If they do, then they are P1, because as you yourself pointed out, only one of you can be, and she's paying !