Based on my looking at and recently buying a GMC Sierra, you can do this with a properly equipped 1/2 ton. But I'm talking new or near-new as the 1/2 tons are a different breed than they were just a few years ago.
Couple of things. Ignore the dry weight. And your hitch weight won't be 678 lbs. Use the GVWR of 7678 and work from there. Hitch will be 12-15% of that so worse case is 1,151. My trailer is a GVWR of 8,200 and my hitch weight is 1,050. Sticker say 520 or something stupid.
Truck I bought is a 2015 GMC Sierra Crew Cab short box with the max trailer package. 5.3l, 6 spd and 3.73 gears. Towing capacity is 10,800 and the carrying capacity is 2,015 as per the door sticker. I looked at Fords but could not find one with their max payload package so the carrying capacity was well under the 2K mark. I don't think the Tundra's have the same ratings as the Fords or GM's and with the recent buy-backs by Dodge, I didn't even look at them.
I moved from a 2001 3/4 Diesel so I know I took a bit of a step back in towing ability, but my trailer is within the weight limits and feels comfortable behind the new truck. I wanted new and wasn't paying for a diesel this time around. As I don't tow a lot, I also wanted to stay with a 1/2 ton. The 3/4 tons would have been a fair bit more money as they did not have the same deals on them.
You can't do anything to a 1/2 ton to make it carry or pull more weight. Air bags etc. don't change the sticker that the DOT looks at if you are stopped. They may make carrying the same weight easier (I'm thinking of adding Timbrens or air bags to mine) but they don't change the actual capacity.
