It can be done with one pump, but not the way you describe. If you use a large stock tank for the fish, be sure to paint the interior white so you can see the fish. I have rubbermaid tanks and it is tough to see the fish.

No reason to have a separate sump. You can put the fish in the bottom and the grow beds above. Flow rate should be about 1 turn over an hour minimum. That means if your tank is 300 gallons, you need a 300 gallon per hour flow rate. Design for double that as things don't always stay optimum. It doesn't hurt the fish to have the water level go up & down so long as they are left with enough water at the low point. If you have many beds, they won't cycle all at the same time, so the variation isn't likely to be all that much.

Grow bed volume should be about 2 times the tank volume.

Make all your plumbing easy to remove & clean. Do it now, or do it later. You'll do it, or give up.