It is very easy to get between the arrivals and departures area in T5. At the AirTrain end of the terminal there are elevators and escalators that can take you up and down. And I'm pretty sure the pet relief area is at that end as well so it is reasonably convenient for that.
I'd probably go to the check-in area, check your bags and then take the pup down stairs for relief before heading back up into the security line and into the terminal. Makes it easier to manage all the bits described.