No, not Cam synchronizer, that squeals, more like slipping fan belt sound
Check WOT(wide open throttle) relay in engine fuse box, also called AC Clutch relay.
When you press gas pedal to the floor the computer thinks you want all available engine power so it will cut power to AC compressor using the WOT relay.
Your sound does sound like a relay "clicking" but in the cab so not sure what that would be.
Just pull out WOT relay and drive truck to see if clicking goes away at WOT
Another place to check for clicking relay is in the Smart Junction Box(SJB), in the 2005 that should be in the cab, passenger side kink panel, remove the panel then test drive to see if you can hear it better