There are many ways to tackle this, depending on the architecture of your system.
Does your internal charging system have any sort of outputs which indicate that it is in charging mode? If so, you could simply take that signal and use it to turn off the power to the motherboard via a relay or solid state relay (SSR).
Does your sphere have to be lined up in a specific orientation to align internal and external coils for charging? If so, you could strategically place a magnet near this interface and place a magnetic sensor, such as a hall sensor or reed switch directly opposite the magnet, internally, which would be triggered when placed in close proximity. The signal from this would be used to turn off power to the mother board with a relay or SSR.
Another possibility would be to place a current sensor between the battery and the charging circuit to detect current flow in that path (making sure it does not sense current going out of the battery to the motherboard) and then using the signal from that to turn off power to the motherboard.
Still another option would be to monitor the voltage coming from the battery charger to the battery and use this to trigger the output relay. To do this, one would have to place a diode between the charging circuit and the battery and place a pull-down resistor (10k would probably work) between the anode and the negative side of the battery so that a significant voltage is measured only when the charger is on. One problem with this approach is that there would be a voltage drop across the diode, which would reduce the voltage applied to the battery.
If the sphere will be subject to lots of bumps and jostling (pretty likely, I would surmise) then using mechanical devices are typically less reliable. In your case, I would lean toward semiconductor-based devices such as Hall sensors and solid state switches over reed switches and mechanical relays.
One thing you will need to determine is how much current your system will draw. You will have to keep that in mind when specifying your relay, as you will want to leave yourself some headroom for current surges and such, which are common in motor control applications.
Here are some categories in which to look for options:
DC capable Solid State Relays (note that the higher current rated SSRs require heat sinking, which could be a problem in your enclosed space)
Current sensing op-amps
More current sensing IC’s