In my experience, most of the thinner sheet stock is not flat - particularly the plywoods. Of course, these are sheets from a home improvement store and not a lumber yard that may have better products.
Yes this is very dependent on the quality of the plywood or other sheet goods. Good plywood is much more likely to leave the factory dead flat and stable compared to the cheaper stuff typically stocked in a big-box store, but in addition they are infamous for their indifferent storage of wood and wood products, virtually guaranteeing that you won't get whatever it is as flat as it could be.
I'm considering baltic birch if I can get it to lay flat in the frame at that size, but I'm not opposed to a masonite sheet if I have to.
Point to remember is that masonite should actually be Masonite with a capital M. Masonite is a brand name that has become semi-synonymous with hardboard (in the US at least) but it's important to know whether you're buying a Masonite product (generally good to excellent quality) or just generic "masonite", which could be pretty indifferent or outright poor.
Anyway that aside, if you can find 1/4" hardboard (specifically S2S tempered hardboard) then I think you'd be happy with its flatness and stability, IF it was stored properly by whoever is selling it. In my experience tempered hardboard is much more likely to be flat than plywoods of the same thickness in roughly the same price bracket, but of course only if during storage it was kept dead flat.
Less desirable IMO but also an option would be MDF. MDF is as the name says a medium-density board so it's not as strong as hardboard (a type of HDF) but it is lighter, which may be desirable here. And well-made MDF is decent stuff, despite how often the product is derided in woodworking forums. This is unfortunately because there is a lot of crappy MDF being sold these days, but like anything a crappy example of something doesn't speak to the quality of the product overall.
Also see my closing comment here about the mixed quality of Baltic birch plywood.