What type of vehicle do I need to tow a 5th wheel with a GVWR of 9650 lbs?

I would like a less powerful/cheaper truck that gets better gas mileage. If a Ford F150 can tow 12,000 lbs, why would this not be adequate? According to the online salesperson, I need a F250 or F350. They say a F150 cannot tow a 5th wheel.
Added (1). So, I could tow a travel trailer weighing 9650lbs, but not a 5th wheel weighing 9650lbs?

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