Hi,
Sorry for the late reply.
The 3.7V battery is not suitable to power the seeeduino v2.2 via external power, because it will first route 5V regulator, then 3.3v regulator, and will be not enough power for seeeduino v2.2.
Maybe you can use this method:
First, switch the switch to 3.3V.
Second, connect the positive of the battery to the 3.3V pin of the seeeduino v2.2, and the negative of the battery to the GND of seeeduino v2.2.
So the battery can supply 3.3V power for the seeeduino, and it will work well.
Regards.