Yeah, you could sponge the surface. Brushing would work as well. With media that can be rolled up and are sufficiently stiff to retain integrity (so not the washi, the canvas may work), you can also process them in a trough (a length of rain gutter with end caps for instance). Another option would be to spray the solutions onto the surface using a spray bottle. Evidently, wear a mask and protect your skin from the alkaline developer, and I'd use a spray bottle as opposed to a misting bottle to reduce the risk of breathing the solutions.
How to get even application with a sponge etc?
Yes, that's a concern, as would the potential for damage since you're working with a delicate emulsion. Definitely use hardener in the emulsion, that's for sure. Evenness is perhaps less of an issue than it may appear beforehand; the reason is you can essentially develop out the image, so any areas that are slow to catch up, will eventually catch up for the most part. Whether it'll be perfect, IDK, but I guess that the texture of the materials you propose to use will hide some of the potential issues anyway.
I'd just try some small sections to figure out methods that work well.