Wiring low voltage doorbell wire through exterior wall and studs
Your plan has a good chance of succeeding as I have done similar work and was successful. I would suggest adding a new junction box, old work box, close to your existing outlet. You can't just go around eliminating outlets because you need the box. Code requires where you need to put them. Just add a new box, pigtail the feed from the existing outlet to the new box and install your transformer. Getting your new wire down there before installing the new box will make it easier.