Powering priority is:
1- coaxial input
2- Hirose input
3- internal batteries
Coaxial input takes priority if power is present through it.
If coaxial input power fails, recorder switches to Hirose input if present, if not it switches to internal batteries.
If Hirose input power fails, recorder switches to internal batteries.
If using both the coaxial and Hirose inputs, power will be drawn from the Hirose input until the coaxial input fails. < Does the Tekkeon battery shut off if no power-draw is sensed? If so it's not suitable for use as a secondary back up battery into the Hirose input for that reason.
This is why I prefer external batteries that use a latching on/off switch (battery stays 'on' until switched 'off' regardless of power-draw or not) instead of a momentary on/off switch. A momentary on/off switch often indicates that the battery features a power-saving mode and will shut itself down if no draw is sensed after an amount of time.
If that is the case, the work around may be running the two in parallel into the same power input. In that case each battery should then provide half the current drawn by the recorder.