I would consider some kind of covered platform that you can place bedding - shavings is what I would use - that you can easily scrape out and replace. A not-too-steep ramp up to the platform covered with plastic poultry fence for traction would be good.
Ordinarily, when someone has a muddy run, I recommend making a shallow channel on the downhill side to direct water into a garden or depression with mulch (planting something like shrubby willow - not near the septic! - or other perennial, perhaps a flowering perennial shrub).
But for this atypical monsoon season, I would consider a simple structure. A wooden pallet covered with plastic poultry fence ought to suit them - and a bit of roofing material (I think it's made of fiberglass or polycarbonate) seems pretty straightforward. And no, there is no guarantee they will love it. But they might.
If there is mud that doesn't dry up, it can begin to breed too many bacteria especially in warm weather, and that can lead to bumblefoot and other problems.