The battery swap is definitely a good first step.
I had this exact issue at the dealer. I had ordered a brand new key to be cut, but when it arrived they couldn't get it to open my trunk. After an hour of troubleshooting, they realized the remote half of the key was the wrong part number for my car. Swapping the remote half with the proper part number fixed the issue.
With that said, there is a process you can perform that might fix the issue, but it may require one of your original keys. Do you have one? These are the steps:
1) Unlock the door to your car using the key blade
2) Insert your old key blade into the ignition and turn it to the "on" position, but do not start the car.
3) Roll down your driver's side window
4) Step out of the car and insert your new key into the driver's door lock, now turn the key to the left and hold it in that position for 10 seconds.
5) Now press the lock button on the FOB
6) Now press the unlock button on the FOB
The only other process I know of to pair keys requires VCDS or VagTacho.