1. If you drill through the joist you might as well cut it off, because that is what the hole would do anway.
2. If that is a 2x4 wall a 3" fitting would be about 3 3/4" o.d., completely filling the cavity, and still project outside the stud, which does not appear to be the case in that picture.
3. I would opt for a different method, but would have to be there to see what is possible.