If the silt is soft, you may be able to use a dredge on a boom, and vacuum the silt off the bottom, pump it 1000 feet away, and distribute.
12 acres, 2 feet deep is a million cubic feet = 35000 cubic yards. That's a lot of dirt.
A track hoe with a 2 yard bucket will take 18000 pick up and swings. AT 20 seconds each, that's 6000 minutes -- 100 hours. Unless the lake is awfully skinny, the middle of it is at least 300 feet from the nearest bank. So unless you get the bottom solid enough to drive the truck on, it will take several moves with the track hoe to get the dirt to shore. Change that 100 to 600 hours, assuming an average of 6 moves to get it to the edge.
If you are hauling with a dump truck, it's 3500 10 yard loads, or 1500 25 yard loads.
And this assumes that you are only taking out 2 feet. More better is to make it deep enough that it won't need to be done soon. Here, the magic number is usually 15 feet so that trout can overwinter.
This project is not trivial. The ONLY way I can see this working is if you can establish a market for the silt.