I tow a 2002 Grand Vitara 4X4 with Auto. Have been towing it since new, has 67000 on the speedo now, not including the tow miles which do not register. I have towed it as many as 500 miles per day cross country with no run up at 200 miles, no problems. I never run up engine, and have never had a battery problem, not much is powered up in the accessory position of ignition I suppose. I usually tow at just below 60 MPH, but have towed faster. My advise, tow as fast as you are comfortable with, and as far as you like, and enjoy the Suzy. They tow really well.
I towed ours from CA to Wash DC and back.