As whopity says, I'd log the flights, but make it clear that it was for interest only and not licencing purposes. I usually do this by entering take off and landing times, but not putting a total time in any column, thereby totals when I add up the flights are unaffected. In remarks, I also put for interest only.
People get very wound up over what you can and can't log, the facts of it are there is only one PIC, and you need to decide that before you go (whether you have a license or not) and there are more reasons to log things than just for licence issue, you just need to be very clear what will count towards your PPL when you apply.