You might try DL in the bottom of that setup but I'd not even bother with it in the top....too small for any real purpose.
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You can sprinkle in some good garden soil (nothing with artificial fertilizers) to help your DLM inside your elevated coop get inoculated with the soil's beneficial microbes.
I add right on top when I add new bedding. This will bury and trap any moisture in the previous layer so that it can help with composting the litter pack. I don't ever stir my litter, though when I started out with DL I encouraged the chickens to do it....but that was before I really understood the capabilities of a true deep litter and had only a deep bedding system with the use of pine shavings.Do you add right on top or must it all be stirred and turned over first?