Well, depending on how much soil we're talking about, adding plants might help. I've found that bare root plants will attract an amazing amount of dirt to themselves, creating their own root ball of sorts.

And while dirt will eventually settle, you many not be happy knowing it will get stirred up any time you stir the water up. You could filter it out - letting the water flow through ordinary quilt batting would help.

I second the quilt batting recommedation.

Also water clarifier. It causes the particulates to clump and sink.

There's no question to me that draining and refilling is hands down the way to go. Pump it out and scoop and shop vac the last bit and refill. Done in an afternoon.

For soil to cloud water it has to be extremely fine. Imagine soil that is so small it doesn't sink. That's small. Filtering isn't an option imo.