As you intend to paint the finished frame, I would, without doubt rout the whole thing from a sheet of MDF. The fact that you have the template makes this a very simple job. Draw around the frame, inside and out, cut away the waste using whatever means you have, a jig saw is ideal, some waste must be left. Attach the template to the MDF using pins or carpet tape and with a straight bearing cutter clean-up inside and outside. You now have a nice clean frame which just needs some edge treatment using whatever cutter/s you have.
Now don't let anyone tell you that MDF wouldn't survive in such an environment, about eight years ago I made a large bathroom cabinet, about 3ft. wide, I sealed it, under coated it followed by several sprayed coats of high gloss lacquer. It is still as good as the day I made it.
Nothing but heaven itself is better than a friend who is really a friend. - Plautus