I thought I'd post an update. I hooked our Fluval 407 (canister filter) back up and added to the tank a Minijet 606 pump we already had. The Fluval's intake arm extends down to a few inches above the sand bed. Between that and the Pro-Clear 150 drawing water from the top, it's gotten rid of a good deal of the detritus the Minijet pushes around.
Thanks again for the advice, guys. I wish I'd taken a better before picture so I could post it beside the result for a comparison. In any event, I'm much happier with the tank now.