Conclusion:
The rig pictured ran for a few hours, longer than I expected, and abruptly cut out. The battery showed no voltage and would not even light a 2 mA LED.
I applied the charger for a few seconds and the battery came back to life, showing 3.6 volts. When the 5 ohm load was applied it quickly sank to 2.9 volts and then cut out.
It looks like I can safely discharge the battery until it cuts out and then connect it to the prototype. When the users receive it, they will plug it in to the USB supply and the Notecarrier, Swan etc. will start up within a few seconds.
I can buy a suitable PKCELL 2,000 mAH cell from a local supplier in the UK. It would be much easier in the US and EU.
Rob