Powering MDBT42Q over battery connected to recharger?

Is it possible to power the board over the recharger?

Depends on the recharger. A good recharger detects the battery voltage and polarity and won't provide any power to a dead battery (0 V) or with wrong polarity (negative). It seems you have a good recharger.

When I connect it to the Bat and GND pin, it seems to work. Is this the right connection?

Yes. There is the risk of overloading your charger if your circuit draws more current than the charger can provide. If you connect only the MDBT42Q it'll be fine, but if you add a motor or 100 neopixels you might eventually overload the charger once the battery is drained.