In the situation you describe, you're not really having an external supervisor. You're a PhD candidate in a university and your research happens to be conducted in a Fachhochschule. The central issue as you said is that the Fachhochschule isn't accredited to deliver doctoral degrees.
So what you are asking is: can I join a graduate program in a US university and conduct research in a German Fachhochschule.
I'm not saying it's impossible, but it's seems like a very long shot.
- First you will need to be accepted in a graduate program with (presumably) a Bachelor from a Fachhochschule. I think just that is going to be a major obstacle, especially if you're aiming at a prestigious institution (see: Is it realistic to apply to a Master at MIT, Harvard, Yale, Berkeley, Stanford with a Bachelor of a German Fachhochschule?)
- Then IF that works out the Fachhochschule’s requirements will be irrelevant. You will in many cases have courses requirements, have to pass quals exams, and all other requirements from the US graduate program. Hardly compatible with you living on another continent.
- The research typically conducted in Fachhoschulen can be very different (typically more applied or translational) than what is done in universities. It might be hard to meet the university's standards for a PhD thesis.
Specifically to your question "will the student have to pay regular fees / tuition even if it is "thesis only"?" the answer is quite probably yes. Usually graduate students and their supervisors seek funding to cover all costs including tuition and stipend.
Maybe you should re-think your strategy and possible try to do a Masters in a German University first. Then applying to a US graduate program might be more plausible. Alternatively you could look at having co-supervision with a professor in a local university.