In my climate, plastic has such a short lifespan, I wouldn't use adhesive because it would be a nuisance to get off when the plastic needs replacing.
I would use cedar lath and stainless steel screws and sandwich the plastic between the
wood lid and the lath, leaving a couple of 1" gaps near each bottom corner for rain that lands on the lid to drain off.
Have you already got hinges? If not, I would also look for hinges that have pins so you can remove the lid by simply pulling the pins. This will make it easy to store the lids out of the sun when they're not needed, thus extending the life of the plastic.
Plastic has lots of benefits, but it's also a pollutant, so as much as I'm not ready to live without it, I keep trying harder and harder to use it carefully.
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