Since this is the part of the door that gets the most stress, I'd actually advise not using epoxy or CA glues for this. Epoxy will be harder to get into the voids, and CA glue will either soak into the open pores of the typical softwood construction of these doors or be so thick and fast-setting it will be a bear to work into the split. It'll be pricey, too.
Use a clamp as a spreader and gently open the split up and then use a syringe and/or a thin glue spatula to work as much wood glue into the void as you can. You really want to work as much as you can into the split, working the spatula as deep as you can go. The idea is that you want a fair amount of messy squeeze-out once you clamp it up. Keep adding glue and taking the pressure off the spreader occasionally until you see squeeze-out all along the split.
Be generous with the glue, and take your time. Those little cheap suction cups used to hang doodads on window panes can be used along the split to force glue deep into the void.
Then remove the spreader and clamp the entire stile up firmly. There should be a fair amount of squeeze-out that you can deal with in the usual manner.
Use a wood glue that has a longer set-up time so you can take your time jamming it in there.
You can take a cue from furniture and guitar repair folks and first drill small holes at the ends of the split to try and keep it from lengthening as the door is stressed.
You may want to cleanup the edges of the void where the plate goes in with a router or chisel so the hardware isn't trying to force the same split to open.