Can you power the ESP8266E-12 with any of the 3.3V pins?
On the nodeMCU board all the 3.3v pins have continuity. That means they all do the same thing. And yes, you can apply ~3.3v to them. I have trouble with voltages under ~2.8 though, 2.5v is likely not going to work.
Use a buck+boost to keep it 3.3v if needed. For cheap dc/dc converters, I strongly prefer ones with XL semi chips; clean enough to please the ESPs.