I think Willys is right about the quality of plywood in general these days. I hope to get around the problem, at least partially, by the fact that my bed is fairly short (54"), and I'm going to use a solid core plywood which tends to keep its shape a little better. (The drawback to solid core is that it is not glued for external use - which I hope to counter by polyurethaning all the edges as well as the flat surfaces). I think I can get away with 5/8" going this route. or possibly even 1/2"
I'm also considering going all the way down to 1/8" or 1/4" for the floor. I figure the floor really doesn't need to hold any weight since it is supported by the real floor right below. Basically this piece of 1/8" will just act to keep the sides and ends square and to have something to attach the upholstery or carpet to.
Another thought has been bouncing around in my head as well and that is to use something like hollow core doors for the sides. In the past I've altered hollow core doors and made some from scratch. They tend to be very straight and they are light weight. It's a lot of work to go to. But if one covered them totally with upholstery you wouldn't have to be that meticulous during their construction.
One final option I've considered is to make all the piece of the bed liner independent - so it can be taken out piece by piece - and basically holds itself in place with a few brackets or snaps. This doesn't get to your total weight problem, but it gets more to my "ease of handling" concern.
But I'm still a little like you - looking for the perfect, lightweight material that will keep its shape and not scratch or mar the sides or floor of the actual bed.