I would say that if you are flying as Pi/c, then you log the hours. That's not rocket science.
However.......... Why can't you sign the tech log? Who is it that you are hiding the full details of the flight from?
That seems a bit iffy to me. Who else can sign it for that flight? I take it the tech log has a box for the Pi/c to sign. Why on earth would anyone else sign for it?
Maybe you need to think on the bigger picture here. When the owner is suggesting that you do something that's clearly not right, how's the maintenance working out for that aircraft? Maybe you are not the only person not signing the tech log, maybe there are more pages than you know about. Maybe it's not about the signature, maybe he just doesn't want the hours recorded.
Remember, if there's any doubt, there's no doubt.