Note on wooden materials. Caution to those using engineered panels like oriented strand board (OSB), MDF, and pressure treated (with ACQ) wood available from regular home improvement places.
OSB is not good to use in exterior applications because it tends to swell when it absorbs water then breaks apart. Don't believe me? Put a piece in a bucket of water and see for yourself! This is why we cover our homes with breathable water resistant membranes over it, something to consider in a project like this as OSB is great if kept dry. For this reason water absorption reason I would avoid using MDF as well.
Plywood has many advantages over a solid piece of wood because the thicker you go with it the more laminated pieces of wood are changing directions. This is why it is more resistant to warping than a solid piece of wood, all those different layers keep each other straight as wood will warp in one direction and since they're "fighting one another" they keep each other straight. However if left untreated/sealed it will rot on you. That being said one could presume pressure treated (ACQ) plywood would be ideal, but I would not recommend it in our tubs. The ACQ preservative is very corrosive to many metals so unless you've bedlinered your tub, use ACQ proof fasteners, and throw caution to the wind, I would avoid it like the plague for this project. (read this
for info on ACQ and OK fasteners)
That being said, there are some marine grade plywood a that would do well or on the cheap you could just use an exterior paint on regular plywood for this job before wrapping it with carpet or just leaving it as is and just replace it if it ever rots out on you.
Then again if you bedlinered the wood before you fasten it on, you could probably get away with any option as it would be encapsulated with a rubberized or epoxy layer. Even so, I would still avoid pressure treated, it weighs a lot in comparison!
EDIT: just watched a couple videos
on plywood testing, regular vs marine, as well as adhesive testing, and I can save you some time by stating don't waste your money on marine, just get the regular stuff and coat it in epoxy and it should rock in our application with/without carpet