I've never made a repair such as this. But will offer a suggestion none-the-less.
Quote:
The seat is comprised of 3- 8" wide x 1/2 boards.
How is the under structure built?
Is it your intent to close the cracks or simply prevent further opening?
I would think with material that thin that you could pinch the crack back together with clamps, route the butterfly all the way through (with a backer as suggested) and then glue it back together with the butterfly in place.
I suspect the real trick for an outdoor project is using a weather proof (or resistant) glue and butterfly.
Epoxy? Gorilla Glue? TBIII?