Taking a page from Rick Baker's policy not to have a "magician's code", I want to share a "magic trick".
The brush is a silicone paddle from a frosting kit at Michael's. It was a pack of 3, and a little expensive, but these things last and are easy to clean off with rubbing alcohol. The test piece is a strip of self-adhering latex bandage.
The neat thing about using the paddle is that the only texture is in the material itself; if the surface is smooth to begin with, or if you use the paddle to apply the liquid latex, it basically acts like a skweegie, keeping the surface relief low. I'll have to experiment more with this, but the idea is that you can PAX-paint a puppet without sponging it on, which results in a smoother surface with no added texture. Another possiblity is to wear nitrile gloves and dip a finger in the PAX, and use that like a brush. You will be able to paint the puppet without leaving any fingerprints.
Whoops, forgot the most important part: The reason for not using a paint brush is that PAX and plain liquid latex will clog the bristles. Anyway...
Clog the bristles? Well, that's a very euphemistic and rather misleading way of saying it will completely and utterly destroy a brush, turning it into a solid lump of rubber!
I was trying to put it nicely. :)
Someone had a concern that the silicone paddle would grab on the paint and tear it off. To prevent that, you just dip it in water before the PAX, and that would lubricate the last layer enough to get more on without disturbing what's already there. Rubber is slippery when wet and it cures through evaporation, so that would just make sense. I'm curious if it's a good idea or bad idea, though. No one has said either way.