The face is very thin, would need very small dowels.If its a clean break & it fits together nicely, I would just glue & clamp it. If you're really worried about it breaking again you could peg it with a few dowels.
Any suggestions on type of glue or epoxy to use? The screws would probably split the wood, not sure. The floor seams are not perforated, it is because of the lighting and camera effects.I believe I would clean as much of the old glue off as you can and put the piece back on with two part epoxy. I would probably also drill a couple of holes and put some screws in it from the underside.....Curious about the floor. Why does the seams look perforated?
It is an old clock, maybe 1940's, but I don't think is valuable because I heard it was built from a kit.I would not use any screw or dowels on this piece myself.You did not say if this was a antique or not.I would like to know the age as that has a lot to do with the way about doing the repair.Judging ny the pics it looks like an old clock and the wrong type of repair could diminish the value considerably.