Joe,
This is spectacular. Also, I updated my previous comment just now, before I saw yours. I mentioned there that I was pretty sure you were familiar with the academic side of these controversies. I'm more sure now.
I agree with you, generally. At that point what we are talking about becomes a form of communication that is mediated through body senses (probably touch and proprioception). It would be a very slight bit like the mentalists I discussed earlier in the story, only without the need for the actual physical contact - any physical interaction would be mediated through electricity.
I think if we were to demonstrate something like that, it would fall quite easily into the realm of natural sensory perceptions that we are presently ignorant about. It wouldn't be quite the same as the brain-to-brain requirement baked into the definition of telepathy. You would essentially be saying that humans can garner information through their body senses, and while scientists would be skeptical and demand a pretty high standard of evidence that information was being transmitted electromagnetically, I can't imagine that they would wholeheartedly array themselves against it they way that they do with anything related to true PSI.
On that note, there are other subtle forms of touch-based communication that tend to go unnoticed by humans, as well. It wouldn't surprise me, for example, to find that many forms of communication between groups of animals are touch-based in nature. If I recall ,there is evidence that the pads of elephants feet are so sensitive to vibration on the ground that they can gather rough information on other elephants in the area from vibration alone. Birds, dolphins, and fish might be similar - I think, for example, that we fail to appreciate the effect a flock of birds has on the air they occupy--and the incredible amout of physical, tactile information that might convey to individual members of the flock about things like which direction to go, how fast to go, etc...
There is one challenge to the idea of a pseudo-telepathy based on electromagnetic fields that I think would need to be resolved, however, and that is the nature of the information communicated. Building a 'bridge" between two organisms (say, a human and a dolphin) electromagnetically would not mean that the information perceived on the human side of things is in any way congruent with the state of the animal on the other end. Stand close enough to another person and you may feel the warmth they radiate but that says little about what they think of you. And unfortunately this means that a great deal of what humans claim to "feel" on their end would still need to be studied with skepticism.
I suppose there's a lot of room in there for learning? Maybe you learn that feeling X in yourself corresponds to state Y in whatever organism you are claiming a link with.
Regarding relying on phenomenological experience; it makes a lot of sense as a vehicle for hypothesis generation - but it would be exceptionally challenging, as you mention, to come up with a testable hypothesis that could properly falsify the notion of a body-mediated form of telepathy.
It's certainly worth thinking about, though. It seems the logistics of the first, most basic aspect would be simple enough - break the electromagnetic link and see what changes.
But that's where all the difficulties come in. You'd have to have a way of engineering a situation where that variable was isolated, which seems tough. And you'd have to have a way of measuring the changes that was unimpeachable. As you said; you'd need an objective means of testing it.
A reasonable first step, though, would be to demonstrate that there are state-dependent changes in the electromagnetic properties of an organism that can be detected with scientific instruments. And before that you would need to determine if there are any electromagnetic properties at all which could be changed. I'd need to research further before even being able to speculate about that except in the broadest sense.