Since no one else is going to bite, I'll throw in half a cent worth.

No matter what method you do, nothing is going to completely turn that jungle into what you want on the first go. Yes, it will grow back. If it were me, I would hire folks to clear it down to the first including stump removal and chasing the roots, and then keep it mowed. Come this fall, seed with clover or grass or whatever your preferred groundcover is and start mowing with the mower blade up high to help the grasses compete with the weeds choke out the weeds.

Somewhere in there you may also want to get a soil test so that you can take measures to optimize the pH for what you are wanting to grow.

Trust me...anyone you hire for that little amount (even $2400.) is not going to be gentle around the roots of the trees you want to keep.
Better to have the treedroppers take the weed trees down leaving a three foot stump that a carefully watched back-hoe operator can grasp and pull out the stumps roots and all while keeping the bulk of machine weight off the root areas of trees you're leaving.
Leaving the stumps simply ground down leaves food for termites and fungi.

Rather than bring in equipment to remove stumps, just paint a ring of herbicide around the outer edge of remaining stumps. Less disruption to the rest of the area and the roots of the dying trees can hold the soil for a little while to minimize any erosion. Any pine trees, by the way, don't need herbicide, they will not grow back.

And I agree, it won't be perfect in just one attempt. Once you clear out the jungle, new sunlight in the area will allow seeds on the ground to sprout and you'll need to manage those.

One of South Carolina's several nicknames is "the Swamp State." If your backyard is plagued with mosquitoes, you may have one of Edgar Allan Poe's infamous tarns hidden away, or maybe even a swamp, in which your mosquitoes are breeding. Drain the swamp, and you'll have fewer mosquitoes.

But seriously, you should subscribe to the print or online version of _The South Carolina Market Bulletin_ in which you'll find a veritable gold mine of Carolinians advertising, among many other things, their skills at land clearance. _The Market Bulletin_ is a fortnightly publication of the South Carolina Department of Agriculture and may be had for ten dollars per year--a steal. You would also be wise to consult the local office of the Clemson University Extension Service. A part of their job is to assist Carolinians with problems like yours. I'm a horticultural fool who prefers to let jungle be jungle. It has a certain mystique, especially if there's a hidden tarn to be enjoyed.

Put in a fence and buy a couple of goats. When the goats have the brambles to where you want them have a BBQ.

Yankee Dog

Cut the trees down & drill 3/8 to1/2 inch holes in the stumps about 2 inches deep & fill them with table salt up to 5/16th, then fill to the top with hot candle wax.
The tree stump will die & rot.
Goats will take care of the rest of the weeds & new growth.
It is the cheaper way & some compost.

I know this is a day late and a dollar short......

Holy cow! Someone wanted up to $2400 to clear a 15 ft x 40 ft spot? Were they going to gold-plate the ground before they left?

Maybe you meant 150 ft x 400 ft, because in the smaller space you stated, I don't even see how you could have that many large trees growing.

If I lived closer I would come and help you clear this small area for free. Sounds like someone was trying to rip you off...... I hate it when people get taken advantage of like that.

From experience I can tell you the goat idea is a good one. Some cheap and easy electric fence and 2 good goats ( no Billies ) and you are good to go.