Starting from the floor…
You have your floor. Then you lay down a protective layer to keep your floor from getting dirty. That can be something disposable like plastic bags, contractor bags, plastic shower curtain, a sheet of coroplast, or if you want something more long term, the vinyl I was talking about or even real vinyl remnant flooring from a flooring or home improvement store. Someone also recently mentioned the idea of one of those carpet protector pads you put under an office chair so it can roll better. That could work well for you too. One top of any of those would be the absorbent layer, which would be old towels or uhaul pads and on top of that would be the fleece. The fleece doesn't absorb liquid, it allows liquid to pass thru it and dries quickly. All the pee is then absorbed by the next layer (either uhaul pads or old towels).
So:
Piggy goes on top of the Fleece,
Fleece goes on top of the absorbent layer,
Absorbant layer goes on top of protective layer,
protective layer goes on top of Floor
The vinyl flooring, plastic office chair protective mat or vinyl by the yard from Joanns are all very thick.. so they lay nice and flat and doesn't crinkle up or tear, it's a more long term solution. I bet if you called a couple of flooring stores and asked if they had any scraps/remnants of flexible vinyl flooring they probably would give or sell you a piece.