Clover would work best stays short and adds nitrogen to the soil. We do it in larger settings (fields) should work the same in small areas white clover I believe would give you the best high traffic results
I cleaned out everything else cause I have to replace the plastic. Dropped a few leftover seeds and the entire inside floor and spreading our with somekind of squash . I also had chickens in it in spring so that soil is chock full of goodness.
Growing “grass” in my greenhouse is illegal. And it gets all bushy and tall and competes with all my other plants. Then it’s gets pretty smelly little clusters on it. I’d go with landscape fabric.