well, if the operating voltage is 2.5 to 5.5 volts, then I'd use an adapter between 2.5 and 5 volts. Is there a regulator on the board to reduce the voltage?
What current does this require?
It's likely that if the board has a position for the power socket then it disconnects battery power once the plug is inserted, so one of those extra connections may be for the normally closed contacts in the socket.
Considering that the socket isn't installed, there is either a link somewhere, or the board was not designed for dual power.