I towed a 2010 GT 6M for over 25,000 miles.
If your battery is going flat, I suspect that you are towing your 2010 with the ignition in either the ON or ACC position.
The 2010 HAS a steering wheel lock so you cannot tow with the ignition in the OFF position.
You have a couple of options.
1) Disable the steering wheel lock as I did with my 2010.
HERE is my web page with the How-To details.2) Use the Valet/Emergency key that is in your key fob:
a) Remove the knob that you normally turn to turn ON and START the car.
b) There is a small round button at the base of the knob.
c) Push the button and pull OUT on the knob, away from the steering column.
d) Remove the key from the key fob and insert it into the ignition switch keyhole that was uncovered by removing the knob.
e) Turn the key to the ON position. It is OK to also START the car.
f) Turn the key to the OFF position,
BUT DO NOT REMOVE THE KEY FROM THE IGNITION SWITCH. It is the act of removing the key that actuates the steering wheel lock.
3) Purchase and have cut another key (NOT KEY FOB just the key blank) and use it as in 2) above.
Any of these processes will let you tow with the key in the OFF position, yet with the steering wheel lock UNLOCKED.
Tim