I don't actually have a roofed run...as I have no run at all, but use the DL in my coop.
And I allow rain and melting snow into the coop to get good moisture into the leaves and material to be found there...can't compost without the moisture.
As for the spores, they are present in nature and in any composting mass of materials. They cannot be escaped, no matter what the vet says, as they grow in soil as well. As long as your run as good airflow through it, they won't cause a bit of problem. The only time mold spores become a danger is when the birds are enclosed in an area with mold spores present and the air is continually stale~such as in a house or in coops that are not well ventilated. In and open run, it's much like as if the chickens were just living on a forest floor.
Deep litter has to be deep, so if you merely had a few inches of wet, matted leaves in the run, I could see where it would turn out to be a nasty mess. Try mixing different particle sizes, throwing in the bark, twigs, woody plant stems and anything else that can decompose at different rates and still provide air pockets in the litter. If built deeply enough~even on the stone floor~it should start to work well, while wicking the moisture into the bottom layers and leaving the top fairly springy and merely moist.
You have a small run, so you might want to either flip the manure under the bedding weekly~just a shallow flip of the top layer~ or put a new layer of dry leaves or material on top. It's like lasagna gardening, so think of layering and keeping the manure in the layers. Once you get it deep enough and with varied materials, the chickens will do much of that for you.
Either way you do it, decomposing plant material is going to have any number of mold/fungi spores in it, breaking it down. As long as you provide fresh air flow into that space, your birds shouldn't suffer one bit.