I wouldn't bother. It appears to be split only at the top. The glue will be stronger than the wood. With the door dismounted, dryfit the sood in place to assure a smooth fit. Razor knife or carve away anybumps that may hinder a smooth fit. Not too much, as you don't want the glue to fill voids. Once you're comfy with the way it looks, smear the wood glue on both sides, wait a minute or so and set it in place and tack. Wipe off any excess and paint. The longer screws will attach themselves to the 2x4's that are behind the face board.
The presumption now..it will be strong enough to support most any kid swinging on it again....unless the screws pull out of the door edge!
