Richard J Kinch wrote:
Therefore, I want to know how it was installed in the first place.

Usual method is just sheet metal screws through cleats on the cooler to the
plywood deck.

Yes, you are exactly right. Last weekend, I removed the seat and found
that the seat was mounted on the deck by large screws. The screws were
screwed into wood framing, not into core. I guess this is OK. This
means I don't need to replace the screws with through-bolt (I cannot
anyway). All I need to do to improve upon the existing mounting method
is to replace the core material around the screws with epoxy bedding.

Jay Chan