I'm working on a new project and need advice about laminating purple heart to plywood.

The particular piece I'm working on is appx. 11 x 18 inches. Only one side will be exposed. I have glued up two pieces of PH into a rough size of 12 x 20 x 3/4. The glue up is perfectly flat and the glue joint is undetectable. The finished piece of PH will be set into a rabbet or a dado in a "picture frame" so to speak. This will be the background of this piece.

I was considering resawing the PH on the bandsaw into two 1/4 inch thick pieces (then I'd have two pieces, one for a future project). If I decide to go this route and laminate one of the pieces of PH to a piece of 1/2 or 3/4 birch plywood is seasonal movement of the PH apt to be a concern, causing it to split? Keep in mind that the whole thing will be "floating" in a rabbet or a dado so there will be space provided for movement there. My concern is the movement of the PH after it's glued to the birch plywood. What's the best way to glue this PH to the plywood to avoid a future split in the PH?

Thanks for your thoughts and input.