I despise the LP-E6 family of batteries (I have a couple of EOS 5DS R bodies that use them) and they are the only Canon batteries I have had that failed to perform in a reasonable fashion. The typical failure is a nearly new battery will refuse to charge even though it was pulled from the camera with half or more charge remaining. None of my 1 series cameras have ever had this sort of battery issue.
A different and known good battery is the first thing to try if you haven't already. The older LP-E6 charger may not be as sensitive as later production so they may charge a "bad" battery and show a full charge for a battery that has failed in terms of delivering any reasonable level of current.
These batteries have two sets of contacts, one set for powering the camera and the other are the sense/info terminals (the smaller pin type). Make sure the large terminals on the battery are clean. It is also possible that your in camera battery contacts are dirty but if you clean them, do so very carefully so as not to damage them. A cotton tip with the highest concentrate isopropyl alcohol is useful for this task. A less than perfect contact will allow the low value of current used for charging to flow but at power up your camera is consuming a short lived burst of current as the electronics wake up; if the contacts aren't clean (or the battery is at end of life) then the voltage delivered to the camera will plummet and the camera will show a "dead" battery.
Rodger
EOS 1DX M3, 1DX M2, 1DX, 5DS R, M6 Mark II, 1D M2, EOS 650 (film), many lenses, XF400 video
