You gave a nice description of your installation and what you have measured then posed the question / challenge:
What's happening here? Go for it guys. A real engineering problem.
The basic principles of what is happening are fairly straight forward. I don't mean to be condescending if you already know the following.
The external side of the coax shield is so well isolated (from an RF perspective) from the internal side of the coax shield that it can be considered a "third wire". What happens on this third wire has no meaningful impact on what is happening inside of the coax shield or the center conductor. Sometimes people have problems grasping the third wire concept because their ohm meter tells them otherwise - but this is taking a DC measurement. Because RF current only flows on the skin of any metal conductor, it typically doesn't makes it through to the other side like DC does. Each side of the shield can therefore carry a completely separate signal (frequency / phase / current / etc.). Thus the third wire.
So if you continue to think about it as a third wire, you have one end of it connected near to the feed point of the antenna and the other end is grounded just before it enters your shack. The length of this third wire, even though it is grounded, has an impedance that is allowing RF current to flow along it. This is what you observed with your RF ammeter readings. As you know, it is called Common Mode Current (CMC).
If you lengthened or shortened this third wire (and thus the entire coax) by something over a 1/8 wavelength but less than 1/2 wavelength or you removed the ground from the far end, you would likely observe a different peak current point and level of CMC and you probably would see the SWR of the antenna change. This is because the third wire is carrying some of the current from your transmitter but since it is not a feed line, it is acting as part of the antenna. And as you know, changing the length of any part of an antenna can have a real effect on SWR and other factors.
While CMC is often treated like a roach - something to be eliminated - this is not always the right answer. In the case of some antennas, CMC plays a crucial role in making it work. In these cases, the balancing act is to counteract any ill effects of CMC - such as RF in the shack, RFI in the house, pattern distortion, or noise pickup - but retain the desired role of CMC. Your grounded point just before entering the shack is probably helping to a large degree to keep the CMC out of the shack.
Many portable antennas used by portable QRP operators make good use of CMC and due to the low power and often the remote location, there are negligible bad effects from CMC so remediation is not needed.
You might find it interesting to experiment with removing the balun from your setup. A lot of things could change as a result (for better or worse). Things to check are SWR, RF in the shack, apparent antenna efficiency, noise level pickup, etc. There are enough variables that effect the outcome that a computer model would get you some approximations but field experience is the only way to be certain.
I hope that helps.
- Glenn W9IQ