Hi. More information please. If you can include a pic of your coop run set up, both inside coop and out, that would be helpful. If your coop/run is a stacked affair with coop over all of the run, I advise against DLM. DLM works best in a walk in coop. (preferably with a soil floor, but can be done with a conventional floor) IMO, the run also must be walk in in order to successfully manage DLM b/c of the large input of materials that is required. You will also want to enter the run to harvest the compost, which would be a back breaker with a short run.
Basically, DL is a sheet compost made of varied materials (hay, straw, weeds, aged wood chips, spent coop litter, garden debris, leaves) which the birds benefit from by having a job to do: turning and mixing the materials, inputting their feces. They benefit from: beneficial micro and macro organisms which help to manage the pathogens which would proliferate in a bare soil run. Studies have shown that birds on DL have a healthier digestive and immune system, improved feed conversion rate, decreased disease, improved viability.
As for straw harboring mites, I don't buy that one. Mites are carried by mice, rats, squirrels, birds. A coop can become infested with or without straw. yes, straw can harbor mites. But so can any other material, including pine shavings.
Digging down a foot to provide a basin for the DL: Unless you live in a desert, and plan to line that basin with hardware cloth, I don't advise it. IMO, it would provide more problems than it would solve (including moisture retention) ... UNLESS you intend to dig your basin, leave the coop there on DL for a year or two, then move the coop and use that site for a garden. However, if you do so, I warn you: you should stand back and toss your seeds from a distance. When those seeds hit that fertile bed, they will explode with growth.