If you make the slats narrow and thin enough, you should not have to worry about beveling anything. Narrower slats will go around the curve pretty easily, and the thinner the slats the less the gap between slats will be. At this point, all you would have to do is make the rounded ends to the lid and glue/nail the slats to the ends.
Take your weakness and exploit it. If you feel that you can frame something okay but finishing is not your forte, consider distressing the box before painting or staining it. This can make any of your mistakes appear almost intentional. And, quite honestly, your kid won't care - they will just be impressed that their dad made something for them.
Here's a link to something you might like
http://ana-white.com/2012/04/plans/pirate-toy-chest
