I respectfully disagree that you cannot add more dirt. If by washing down, you mean it will be piled up higher than your door height and so the pile will wash down in front there, simply add railroad ties as a retaining wall across the top of the structure, stretching out on either side, until you reach your desired dirt height and the back fill to them - just as the dummy (er, contractor) did on the walkway to the door. Alternate your joints so it is strong and nail with large spikes. Anchor with rebar.
If it were mine, I would rent a Bobcat, scrape back all the dirt off the top, get 6 inches or more of rigid foamboard across the top and extending a few inches to a foot over on each side (and down the upper portions of the sides if you can) , build my railroad tie retaining wall in front above the doorway, and pile 4 feet of dirt on there. Better yet, get the dummy (er, contractor) to do it in lieu of a lawsuit. And closely watch their every move.
This did not turn out to fit your expected usage nor the contractor's expressed warranty for fitness, so I'd be seeing a lawyer about financial remedies. I have had my share of jack-legs masquerading as craftsmen, and a letter from a lawyer usually is sufficient to get them onboard.
Alternately, you could scrape all the dirt off the top and about a foot on the sides. Then construct a conventional roof on top, using foamboard insulation inside it and Reflectix (
http://www.reflectixinc.com - if you look you will see they have WINE ROOM listed) double-bubble insulation stapled to the rafter with the shiny side UP. Then you put purlins on the rafters and add a reflective metal roof. It turns out so that the Reflectix is on the rafters, then the purlins allow a 1-inch airspace between it and the steel of the roof. AMAZING heat reflective properties. I had this installed on my house to hold down AC bills. Makes a 15-20 degree difference. Of course, I had to stand there and tell the installers how I wanted it done. They had never heard of it! Argued and cussed me. I said, who's paying for the job? That got it done. I already had this on my metal shop building (which I built myself), again a huge cooling effect.
I guess my advice is not to just settle for something inadequate. Your ideas for passive cooling were sound, so get it so they will work!