So I decided to replace it. The resulting tea caddy is pictured below. It's walnut with a plywood bottom. The corners are rabbeted together (a step shaped notch is cut into each side so the ends fit flush with the end of the sides), and the divider is set into a dado (groove) in the ends. And I should mention that all the wood used was scraps from other projects.
The finish is a spray on polyurethane.