Steal 5v from USB Module in 3v Mode
I don't know who designed that board, but it has several questionable connections.

First, CH340C chip datasheet requires V3 pin connected to VCC if VCC is 3.3V. On the board it is permanently connected to decoupling capacitor, which corresponds to 5V configuration.

On the other hand, the LEDs are connected to 3.3V supply, so they will (maybe) operate  in strange mode when the logic levels are 5V.

So basically, the schematics is wrong whether it is configured for 3.3V or 5V power.

I'd recommend leaving the jumper at 3.3V and soldering a wire to the other (5V) pad of the jumper to bring 5V VBUS out of it.

UPDATE:


  
  Set the USB to 5v mode. That requires the power to the ESP8266 use a voltage
  regulator...
  


I realized that the statement above bothered me before. You seem to be under impression that you don't need voltage regulator if USB breakout is set to 3.3V.

However if you power ESP8266 from lithium battery you MUST HAVE voltage regulator between the battery and ESP anyway. Fully charged LiPo can go up to 4.2V, way above 3.6V maxium for ESP.

Also note, that VCC on the USB breakout is an output, not input. It will put out 3.3V when jumper is set to 3.3, but you cannot use it to power ESP, because there is no way to connect battery to the input of on-board AP2112 voltage regulator. Too bad, as it can supply up to 600mA, plenty for both ESP and CH340C.

Well... technically there is a way. You can cut a trace between VBUS of USB jack and input of AP2112, taking care to leave C1 on regulator side and C5 on VBUS side. Then you can solder another wire to "IN" pin of AP2112 and connect it to battery output. Then power ESP from 3.3V VCC output.

So, if you feel adventurous, you can download Eagle files from SparkFun and figure out how to mod the USB breakout. Otherwise you need additional 3.3V regulator between LiPo and ESP.