I would think the mesh with screen over it would work, but a couple points.... maybe: 1. make sure you use a screen that would hold up to water...they will pee on it and you will have to clean it somehow 2. Make sure you trim off all the sharp edges... hardware cloth will cut you up.

You could also consider making the levels out of pvc legs and ceramic tile for tops maybe (attached with gorilla glue or some such. You could still clip fleece over it, but they would be more waterproof to clean / wipe down. I made a platform like that for my BTS and it was easy to clean and seemed pretty stable.

You could try the plastic covering you mentioned... and just see how it goes and replace if needed.

Something else to consider. If this will be inside a cage add-on (wire sides) you could avoid the levels all together and hang hammocks, baskets, etc. instead. Some people will hang small cat-litter trays and fill them with litter to make them easier to clean. I use hammocks made of fleece, which then gets washed every week. I also use climbing ropes.. Large bird rope perches that screw onto bars.