How to screw plywood to a Rubbermaid bin?

Use some washers with the screws and screw through the plastic into the plywood. I don't know how thick your plywood is if it's also 3/8 .... you may need to clip the point of the screw so it doesn't pop through.

Or you can do it your way, which would work also.

gorilla-glue it.  that stuff's amazing.

Lid or no lid, forget screwing anything into that plastic.  Just attach wood strip to the plywood that hold it in place.  If you're smart enough to slap together a waste pump

setup, how is it even possible you can't figure that out?

Find the lid that came with the bin. It CLIPS to the bin. It will be FAR easier to attach the plywood to the lid than to the bin.

if you use rubber washers on your screws, you can screw right through the bottom of the bin into plywood and it won't leak.  try #10 x 5/8 inch wood screws.

Screwing through the plywood, through the bin edge, and into a piece of wood backing up the rim is about as strong a solution as you're going to come up with - sandwiching the rim between two pieces of wood eliminates point loading and brings the entire edge, simultaneously, to bear against shocks and strains.  It will be very strong.

Don't try to hit the plywood from above - drill all the way through the backing piece below, up through the bin edge and through the plywood above.  If your drill bit is only long enough to make a mark on the top plywood's underside and not get all the way through, that's fine - you've made a mark.  Get all the marks done then drill all the way through the top's assorted holes at once, and then you can fasten it down from above.  That the screw holes go all the way through the backing piece under the rim, not important.  It won't affect the holding power at all.

Forget the Rubbermaid and build a box out of plywood. Design the whole thing from scratch. There are various paints and resins that can waterproof the inside after it's made. Your issue is in trying to work with the wrong part to begin with. Stop doing that.

 angle brackets

Put a couple of timber battens (2x1) under the ply and screw through from above and make a snug fit to rim of bucket on long sides, or screw through the wall of the bucket.

I think I would get three pieces of wood, 1x6 probably, and assemble them in a U shape so two of them are against the inside walls of the bin, resting on its bottom, and the third sits on top of them to be flush with the top edge of the bin. Some L-brackets might come in handy. The plywood with the pump can be fastened to the third piece. Fasten the two upright pieces to the sides of the bin from the outside, using screws and the widest washers you can get. A fourth piece, on the floor of the bin between the two uprights, would help make sure they stay put. That one doesn't need to be fastened to the bin, it could even just be jammed in if you cut it just right.