Not a direct answer to your question, but qs an inexpensive option closer to tray development, most home improvement stores sell clear plastic sheet that's intended for use as a vapor barrier. It's pretty thick stuff and you could pretty easily construct a temporary tray without needing to actually store a tray that size.
As far as direct application of developer goes, the only experience I have there is around my attempts at diffusion transfer (ala old school polaroid) where a thickened monobath is rolled out between a negative/positive. I imagine you could add some CMC (Carboxymethylcellulose) to your developer of choice and then apply it with a roller to get good even coverage rapidly. Intuitively that seems more delicate than a brush or sponge, but you'd have to test and see