Will a 2022 Ford F-150 tow a 2027 Forest River Salem FSX 250GT?

No, it's over the limit

Over on payload.

• Tow rating: 325 lb to spare (7,875 lb camper, rating 8,200 lb)
• Payload: Over by 250 lb (1,655 to 1,970 lb used, payload up to 1,405 lb)
• Truck and trailer together: Not sure yet (14,885 lb, GCWR up to 14,650 lb)

Hitch setup

• Weight-distribution hitch (needed): Your camper is over 5,000 lb. Most truck makers require weight distribution here, and many receivers are rated far lower without it. It spreads the tongue weight to your front axle and the camper's axles, so the truck sits level and steers right.
• Sway control (strongly recommended): It stops the camper swinging side to side in crosswinds, when semis pass and on rough roads. At about 33 ft, your camper catches a lot of wind. Many weight-distribution hitches have it built in.
• Spring bars rated about 1,000 to 1,600 lb (size): Match the bars to your loaded tongue weight (945 to 1,260 lb, from an estimate of 12 to 16% of the camper's weight) plus anything heavy behind the truck's rear axle. Bars that are too stiff ride rough. Too light won't level the rig.
• Brake controller (needed): Your camper has electric brakes, and the controller in your cab runs them. Many newer trucks have one built in.
• Measure your tongue weight (tip): We're estimating 945 to 1,260 lb. A tongue weight scale or a hitch with a built-in gauge gives you the real number. Enter it in step 2.

Hitches make towing steadier, but none of them raise your tow rating or payload. The hitch itself counts against payload.