Have you tried with a different, known healthy, battery? I'd first check the frame connection from the battery. Check if the voltage drop is at the battery poles as well as the other components. If ok at the battery but drops along the way, find where the drop starts.
Or could be that something gets hots upon current drain, then expands due to heat and gives a good connection. Any heat along the cabling somewhere?
If you cannot find anything obvious, start an elimination method. F.ex, disconnect the A connection at the regulator to get regulator\dynamo out of the way, short your amperemeter to see if it makes any difference, etc. How is the ammeter behaving during the low voltage period? (maybe not relevant, as you write the horn does not get power either, and if it is standard wiring, the horn does not go trough the amperemeter).
just some ideas to start with. Good Luck, when you find it, tell us what it was!
E.
