First thing I'd say, if you really want to board ASAP, is periodically check back for a new PAT. Sometimes earlier times open up later in the process once people work out their transportation choices.
Next question: Is it a one way rental or week long? If it is a one way rental, you can probably get there even 30-45 minutes prior to your PAT, at the very least this gives you time to drop off your people and return your car so that they won't be waiting for you to check in. There is a chance that if the building is too crowded they won't let you in, but even if your family has to wait outside the building, it still gives you time to drop off the car. If it is a week long rental and you are parking the car there, then 15 minutes before your PAT is probably enough.
As Shmoo said, even if you get through security and check-in, you still have to wait for them to call your boarding number and that is tied to your PAT, not the time you finish check-in.
Still, if you REALLY want to get on ASAP, and you don't mind lines and waiting, you could get to the port earlier so that you will be certain to be finished with check-in and security as soon as your boarding number is called.