You have two sides of the mortice. You have the side that will be covered by the tenon pc and the open visual side. What I might do is on the visual side use a router to make a nice neat clean sided hole, doesn't have to be all that deep, maybe 1/2". You can square up the corners with a chisel. On the other side youcan go the drill bit and chisel route. No need for it to be a clean looking cut as long as the mortise/tenon is pretty well fit it will work fine.