Hi, James,
For the LPCXpresso54114 Board, when you connect the 3.7V Lipo battery to the JP9 header, as you know that the pin2 of JP9 is VDD_LPC5411x node, which provide 3.3V power supply for all the 3.3V chips directly including LPC54114. The VDD voltage of LPC54114 power supply ranges from 1.62 to 3.6V, so 3.7V is a bit high.
BTW, the pin3 of JP9 is an output node of regulator U24 rather than an input, so if you connect 3.7V LIpo battery to JP9, the battery power supply does not pass any regulator.
Whether you can use the 3.7V Lipo battery to power all the 3.3V chips is dependent on if it can provide enough current.
I suggest you use a 5V battery, which can provide at least 100mA current as the USB port can provide, the power is 5V*0.1A=0.5W. You can connect the 5V battery to pin3 of P3, it is okay.
BR
Xiangjun Rong