Well, I guess I'll answer my own question. Found this page:
http://www.harmonicappliances.com/floorboard/floorboard.html
which describes the pin-out of the Line6 Floor Board RJ45 port (compatible with the Flextone I). Assuming the author is correct, a regular POE data/power splitter wouldn't work, because the Flextone output voltage is not what a splitter would expect, and the pins are in the wrong place.