Go with the solid wood if you're more comfortable, the engineering is no big shake.
You are right about 3//8" movement with a lot of domestic hardwoods (oak, cherry, maple, for instance) so go conservative and allow for a half inch. As rich c. said, pin it in the middle, put a couple battens (with slotted holes) on the back to make sure it stays flat. Allow an extra 1/4" in the edge grooves for movement.
Personally, I don't use the balls, but you could work the battens into the frame and use them to keep the panel centered.
Make sure your glue joints are good and tight at the ends just to keep things safe.
On the other hand, if your plywood supplier lets you pick your sheet, flip through a few to see if there is any weird veneer in there. You can field the ply with solid wood or rebate the back to make the face flat. I suppose that's the more Shaker look.
While you are at it, consider PSA backed veneer from a company such as Oakwood. I think you can get it with the thicker backing, or veneer backing, which seem like better quality products. For best results, you have to stick it to a finished surface. I believe 24" is the widest you can get PSA veneer.