3,7V supply circuit as battery
You can supply the device with 3,7V (like the battery) from an external source. The only thing bad is that you have to attach wires to the gold plated battery slots on the device, or you can do it with small crocodile clips to avoid soldering. (better)
If you are ok with this, here is how you do it.
Take off your battery and measure voltage DC with a multimeter or voltmeter between battery leads. Now you know what your battery gives to the device. Example for 3 leads battery: you have 2 positive leads with the ground as reference (one slightly lower than the other) and the actual ground. So you have to supply with 3,7V the same leads that the battery was supplying. You can check while inserting the battery back to the device.
Hot to have the 3,7 Volts supply:
You will need 2 resistors, some capacitors, LM317 regulator, heat sink for that and a higher voltage DC power supply 6-12V.
Get and android device and go to play store. Install "Electrodroid" application. This will help you on sizing the LM317 regulator. Have in mind that this is programmable regulator, so you need 2 resistors to set the output voltage as 3,7V. LM317 is a linear voltage regulator, so it will act as uninterrupted 3,7 Volt battery. Be carefull to get a big heat sink, depending on the current you will be supplying and input voltage, also you can read this device datasheet online.
You can build this circuit on a breadboard if you are familiar with electronics or you can solder point to point the parts, or make a pcb if you can.