Safety first!
Your payload is only 1,000 lbs or so - that's why it's a 1/2 ton truck. If it were a 2500 HD truck I'd say no problem. I've hauled ~1,500 lbs with my '06 1500 for a short distance (about 4-5 miles).
How about borrowing or renting a trailer? 2400 lbs should be well under your max tow rating. It would also be easier to load/unload. Do you have a hitch? Even a bumper hitch will do for this as your tongue weight would only be about 250-300 lbs and a bumper hitch can usually handle up to 500 lb tongue weight.
Do you know what the GVWR is for your truck? It should be on the placard on the door. The payload is the GVWR minus the curb weight of the truck and passengers. For example the GVWR for mine is 6750 lbs. My truck without the driver, passengers and no cargo is 5100 lbs so the net difference is 6700-5100 = 1,600 lbs then subtract my weight (~200 lbs) so in theory I could go as high as 1400 lbs if I carried no passengers. Your truck is probably similar so 2400 lbs would be exceeding max payload by 1,000 lbs or so. Even a short distance with the much overweight is questionable. Remember it's not just the engine and transmission, but brakes and suspension as well. Could it do it? Probably, but you would be well over any safety margins and if there is an accident, you could be cited for being overweight.
If it were me, I'd find a suitable trailer. You'd probably need a double axle trailer and they can be rented from U-Haul for about $20-$30 for the day. 2400 lbs really is pushing it - even for a short distance.