I just tried an AeroPress paper filter on top of the coffee bed, cut so it tightly fit the basket. Some coffee migrated to the top of the basket and some found its way to the bottom surface of the piston. It was easily cleaned off. I will try another shot using now shower screen or filter at all and will post about that here. I'm thinking that if the water is drained well from the coffee cake it won't leave a lot of residue and can be easily wiped off. In the worst case coffee grounds can get into the groove in the piston gasket, and you can use Q tips to clean that.
Trial and error result, at least if not pre-wetting the coffee very slowly: not using any shower screen or filter at all disrupted the coffee cake for a very uneven pull, and I wasn't pouring very fast. A "dispersion screen" is just that. Some kind of barrier really does help.
What I WOULD do for a good cup of coffee!