I think straw would be the best. That is what we use in horse and cattle barns. When the mud dries, the straw will make it even stronger. This is how they made dirt floors in the olden days.
I Have An Outdoor Dog Pen; Grass Is Mostly Gone & Mostly Dirt/mud. What's The Best Ground Cover For Their Pen; Pea Gravel, Sand, Mulch, Even Thought Of Cat Litter.?
We have straw in some places but it gets wet when it rains and I think mold develops in it. Also, bugs of all sorts, I don't like it and that is why I am on here searching like you. Going to pull up straw that is there and then spray with bug killer and keep looking. Our pen is like yours, mud.