Lazor:
My impression is that you are eating a sweet pickle that is going to get a lot more sour. That shallow water is going to become overgrown with weeds which might be great for forage fish to hide, but will eventually cause problems.
My gut instinct is that the most practical (considering cost, etc.) would be to bite the bullet and drain that rascal for a complete overhaul. A few trackhoes (very few) are big enough to effectively reach out far and dig it out; but you are talking about a lot of muck and fairly considerable cost. With one, you might be able to dig one end out deep so the water and fish drain into it while you dig out the rest...
It is impressive that the pond never freezes. Maybe you have a really good spring that circulates well. That spring could be your salvation or could turn into a huge vacuum.
I'm not sure how well the fish would like the muddy water, but I'm sure they'd prefer it to clean air!
Hopefully, the pros will jump in here with a response so I can learn what one really should do in your situation.
Please provide updates periodically on what you do and how it goes.
Jeff
P.S. FWIW: A municipality I know of had their lake drained and dug out. It seems like they dug out a similar depth and size using trackhoe and trucks. Lake was about 6+ acres originally. They did roughly half of it. Seems like cost was around $90K at prevailing wage.