I gots a rubbermaid grow going and I would suspect you could just drill holes in the tub on opposite sides, thread a string through the holes, and use a regular screw and maybe a washer, screwed into the hole with the string to keep it tight, or better yet just make a knot or 2 on the string after it's through the holes to keep it tight. Do it a few times and theres your screen.

But in all honesty, I would think you'd be better off using some heavy LST, especially since you aren't using a second tub for added space. Hmmmm, if your just using a lid, that space is quite small isn't it? If you were gonna use the lid, I would think you'd be better off with something like a tall trash can rather than a storage tub. I don't use a bottom at all for my grow.

Hope this helps.