If you are happy to do some soldering / breadboarding you can probably get this to work by adding a smoothing circuit.
The smoothing circuit will take the PWM output and make it more like a constant voltage level.
This will keep the controller chip happy and allow you to use the PicoBorg normally.
A resistor and a capacitor should be enough to try this.
There is a good explanation of how it all works here: http://provideyourown.com/2011/analogwr ... o-voltage/
To work out the values you want there is a calculator here: http://sim.okawa-denshi.jp/en/PWMtool.php
The frequency used in the examples is 100 Hz.
Vl should be 0.
Vh should be the same as your input (battery) voltage.