I've seen some nice darkrooms, but most darkrooms I've seen don't have a bathroom like yours does, much less a washer and dryer.
Will the sink be used to hold water or just manage water that comes out of the trays (i.e. will there be a plug used in the drain)? I'll assume the later. Also, when you look at sizing the sink, keep in mind you need to make sure you can prevent splashing stop bath into developer.
Given that the base of the sink will be well supported, here's an option: build a sink using 2x4 (or even 5/4x4) cedar and 1/4" cedar plywood. 2x4 cedar for the front and back, plywood for the bottom and ends. Paint the inside with epoxy paint, or even just use spar varnish and silicone caulk the seams. Cedar will cut the weight significantly, though at greater cost. For drainage, simply put a 1x1" piece of wood underneath the end furthest from the drain.
Alternatively, look at the plastic containers used for underbed storage. They may be large enough, at least for 8x10 trays. Consider 6" high divider between developer and stop bath trays. That divider could be anything from 1/4" plywood to varnished cardboard.