RTOM
(Read the owners manual)
If it is rated for 5000 lbs weight carrying and not 5000 lbs with a weight distributing hitch any the load you plan to tow weighs under that you are fine... But keep in mind that max tow rating assumes min weight inside the tow vehicle... Cargo and passengers reduce tow rating.
3 miles at LOW SPEED is even less of a concern...
The initial tow is where you need to be certain...
The dealer is a bonehead tho saying the trans cooler was the only concern... I would be far more concerned with safety and the secondarily concerned with longevity.
That said, I'd say as long as that 5k is a load carrying rating and you don't plan to have 5 fat guys and a week's worth of luggage in the SUV you should be fine.
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