Lots of good suggestion already so I won't repeat, rather I offer a far less craftsmanship alternative - long finish screws. Two in from the stiles one in from the rails. Of course make certain your joint when glued up is tight and clamp a sandwich of wood strips top and bottom face (that still allow you a peak at the joint. These strips will insure that the joint doesn't move as you set the screws. Even with finish screws (I really like GRKs) you should pre bore a pilot hole and for hard wood also slightly enlarge the hole where the screw head will land as the screw head could mushroom or split your wood.
NOW - I would never build cabinet doors this way (I would not typically build mitered corners doors or panels in any event) but if I were repairing one or two that had loose corners that were otherwise solid, or as in this case making up a few doors to match without other tools available, this might be a viable alternative.
"... for when we become in heart completely poor, we at once are the treasurers & disbursers of enormous riches."
WQJudge