A little while ago, I got hold of a SIM800L module and tried to install it in the phone, powered off a 4V supply line from the CM-H333 PCB and...it didn't work. Although the phone was definitely supplying 4V to the module, no flashing lights appeared on the SIM800L.
I suspect the current from this supply line is insufficient to power the (rather current-hungry) GSM module. I think I'll try a different tack - installing a Bluetooth module instead. Although the purist in me would have preferred to get 2G connectivity to the handset (thus resurrecting it as a proper phone) I am happy to settle for shapeshifting the phone into a Bluetooth headset.
The Bluetooth module I'm looking at is the JDY-64 as this includes a microphone input (quite rare among the myriad DIY Bluetooth audio modules on eBay) and it should run off the 4V supply line, requiring a far lower current than the SIM800. Fingers crossed!