As a private pilot wearing my ASIC, am I then allowed to walk up to the RPT aircraft?
You can try it but I would then expect a call from your local state or Fed police officer for a bit of a chat.
Read Walrus' clarification of my post earlier. The ASIC only gives you access insofar as you have a lawful excuse for being airside. Having your 172 parked on the edge of the apron is not an excuse to be up close and personal with the RPT. I'm sure the ramp rats, pilots, refuellers and anyone else around would make sure of that. Despite what many might think, they do know wtf is going on.