JUST-Local has answered your question - to the extent it can be answered.
Since either of you could be P1, the answer is whoever you both agree is P1. And if neither of you agree up front then the issue is likely to be settled by appropriate authorities post accident/claim.
I have no legal knowledge but FWIW I would have thought that signing the techlog is likely to take precedence, over the flight plan. Make sure you're happy with that !
There is a very simple way to clear this up. Ask your client whether they are logging the flight. A good idea to do this before the flight I suggest.