Sorry, for the delay, I had some catch-up to do after the holidays, so my project went on hold.
Steve, the edges of the particle board are clean-cut, so I know the trim will glue well there. However, the edge of the lid is only 1" deep, while the trim I plan to use is 3.5" deep. It will extend 1/4" above the top of the lid, so there will be 2.25" extending below the bottom of the lid. That's why I'm thinking of gluing reinforcements on the bottom of the lid, against that part of the trim. Right now I'm thinking short pieces of a 2"x2" square dowel, spaced periodically along the trim.
But again, the bottom of the lid has one brushed coat of polyurethane on it. That's why I'm wondering if I can glue the supports right to that with wood glue, or if I should try something like Gorilla Glue instead, or try to sand the poly off and go with the wood glue? I'm inclined to think sanding the poly off and using wood glue since that would be a good strong bond, and since Gorilla Glue tends to expand so much as it dries. Does anyone have any advice on this?
George, the lid is two pieces of 1/2" particle board, glued together. You are right, a single piece had a lot of bend to it and would have caused issues. Since the lid needs to lift off, I went with the two pieces of 1/2" glued together - it seems strong and rigid, yet with a reasonable weight.
Thanks again for all your help!