If you've checked and confirmed (best source to my knowledge is handgunlaw.us) that you're legal to have the gun with you loaded in the car in a state-owned rest-stop everywhere but California, I applaud you for the wisdom of checking it out beforehand.
Might be a good idea to download printouts from the site confirming it, not just from handgunlaw.us, but from the "dot-gov" links for each state that you'll find there. It will make things much easier should a discussion of the matter evolve between you and LEOs, or private security, at the given rest stop. Be sure to keep the printouts at hand in the car with you.
In California, I understand that your gun will need to be unloaded, secured out of reach, and separate from ammo. Check with handgunlaw.us for referral on the exact details.
Since you won't have access to a gun, the dog will be your primary line of defense when it's time to close your eyes. You might want to check with California Highway Patrol to see if there are any particular rest stops they recommend for your purpose in terms of safety.
With best wishes for a safe trip,