Upgrade to at least 1/2" plywood. 5/8" or even 3/4" would be better but it may not stay up as easily when you lift the plywood up to get at the storage space under the bed. May need stronger shock struts or to relocate them a bit.
I've seen 2x3 and 1x2 used on the flat to strengthen the underside of the plywood, or even nothing at all. You could also use some angle iron. Another option is to glue & screw another layer of say 3/8" plywood to the underside of what you have now. Just don't extend it to the perimeter so that it will still close properly.
Some units have the bed platform that hinges up at the head of the bed (against the exterior wall). Our first TT was like this. I strengthened the underside of the platform with 2x3 but had to notch the forward side of the underbed storage space wall. The hinges were barely holding onto the wall as well (just into the luan plywood) and I installed a piano hinge instead.
I haven't seen a TT yet where these things aren't flimsy. I'm surprised ours hasn't collapsed when our dog jumps up when we're both on the bed.