Can I use a 3.7V battery for a 3.3V board?

Not for sure, given the many different manufacturers of such boards, and their propensity for substituting in alternative and/or "counterfeit" parts for originally specified components to save on costs. But most 3.3V regulators are "LDO" regulators of some type, one design said it had a rt9193 (which has a 220mV dropout), and the microcontroller will operate below 3.3V anyway (down to 2V if its an actual ST part, which is also in doubt.)