First off, the good news... my sculpture didn't explode or otherwise fall apart. Hurray!
Once I was ready to start painting I took the sculpture outside to begin. I started by laying down layers of dark green and brown on the areas that I wanted to be darker than the rest. This layer of paint would dry for 30 minutes before I added more paint.
Next I layered on the lighter green and scratched back into the surface with steel wool after only five minutes of dry time. It didn't take much to bring the paint right back down to the surface of the sculpture. After experimenting with this for a while I instead sprayed darker green layers carefully on top of the green in the most heavily shadowed areas. I also flipped the chayote upside down to paint the bottom the darker green, increasing the variegated green illusion. To achieve the brown stem and bruised area I sprayed paint onto a q-tip and applied it to the appropriate areas.
Once painted some of the flaws of the piece, like the snout area being a little elongated, didn't seem as bothersome. With color the piece takes on the appearance of a unique chayote rather than a direct representation of the original, while retaining important similarities.