EDIT - Apparently I have a reading comprehension problem. The OP said they were thinking about using the back of the downspout...... Personally I don't have a problem with that. Obviously the cable is going to be exposed and could potentially be tampered with, but I think the odds of that happening are nearly zero.....
Original post:
I would try to run the cable inside the soffit that the gutters are attached to. Generally speaking, those soffits have vents. By accessing the vest and using pull rods to push the cable through (you might need to use several vents along the way), you can probably run that cable pretty easily. Now I realize this is highly dependent on your situation and roof structure, but there is usually a gap between the side of the house and the roof joists. It is certainly something to look into.
Even if you have to push the wire all the way into the attic space, you can then use the attic as a raceway for the cable assuming you have some sort of attic access yourself. Again, using push rods makes this job much easier. Use them to push the wire from the outside into the attic, then along inside the attic, and then again at the other end where you push the rod into the attic and attach the wire to it inside the attic before pulling the rod/wire back out to the external location (assuming that the wire has to go back outside at all).
Example of a push rod.......
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