I would look at the voltage regulator. It sits at the bottom of the frame in between the down tubes. Also, a new battery might not be a bad idea. Ignition on, voltage will drop a little naturally because of the load. Running though, it shouldn't be doing what you are describing. Dave will probably have more knowledge of this and be able to help further but these are my 2 cents for now without pulling out any wiring diagrams. As far as the actual ignition switch not working, that is common. No harm in using a push button directly on the starter. I used one on a custom Softail for many years without a problem.
Welcome to the forum!
-Rohan
Welcome to the forum!
-Rohan