CSD tool shows about -780 mA and the board gets pretty hot but continues to charge. However, I'll probably find somewhere a boost converter which is capable to ouput 1 A. I have charger-only boards from Adafruit too so I'll just need the 3,7 -> 5 V (1A) boost converter.

Power Cell would work if the current could be limited via Jolla but as far as I can see that is not possible. Of course one can use it, but it gets hot and probably has some efficiency problems at those output levels.

Edit: I think this will do: http://www.ebay.com/itm/New-1pcs-1A-...-/191014690726