I do think the DLM is the better litter method. I'm sure others will disagree, too.
Remember that for composting you have to have a moist environment. That's not really what you want in a chicken coop. But, there again, you don't want it to be a dust bowl, either. While it's summer time I'd almost say try sprinkling lightly with water once a week or so and see how it works. Don't get it soggy, of course, maybe just enough to settle the dust a bit and then stir the litter up some.
I don't know what effect wood over the linoleum would have. The wood *could* possibly absorb some moisture and then release it back. Personally I wouldn't go to the trouble.
I don't see why it would hurt to add some of the composting wood chips from the compost pile. Also, if you can get them....leaves!!! That would be a great addition to the chips and other things.
The moisture content seems to be the "biggie" and it will require a sufficient depth of litter for it to hit a "balance" of sorts and to maintain that balance. A shallow layer will dry out quickly and won't have much thermal mass to it. How deep was your litter the first time?
Thanks for sharing your experiences with us. I'm still debating on building my coop with a raised floor or a dirt floor. I have been figuring on the raised floor but some good conversations with some folks (Beekissed, for one of them!) has me considering the dirt floor. At the moment it's a toss-up for me....got to make a decision soon!!!
Best wishes,
Ed