I cut a number of those puzzles last year and gave them away as Christmas presents. I used 1/4" BB, tried 1/8" BB, but the thicker pieces are easier for kids to handle and more durable. I painted with acrylics and finished with spray lacquer. As long as you let the finish cure, no more smell, any finish bought in the US is non toxic.
I stacked cut three pieces at a time and used a #5 SR FD. You could use a #3 but the larger blade makes the puzzle easier to fit together. (wider kerf)
Barry