Great question — answering as the UphillTech team:
If you can't relocate it to either the narrower white bit on the right or one of the red bits, simply use longer screws. A Ring door bell or similar can't weigh more than a few ounces, so you could simply buy longer wood screws with the same type of head that the device came with, carefully drill a clearance hole 1 through the plastic (to ensure you don't crack it), then push the screw through that hole until it hits something solid, then drive the screw into that. Don't drive the screws to the point of deforming the plastic you're mounting it against, but until it's just flush. If you're using a powered driver, leave the screws just short, then use an actual manual screwdriver (I know...
Full answer by FreeMan: https://diy.stackexchange.com/a/209404 (CC BY-SA 4.0).
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