From Espruino's point of view it's not possible, as all the communication is handled by the Bluetooth module itself.
However you might have noticed that the module's pinout looks almost identical to the HC-05. I reckon you could take one of those, unsolder it, and resolder it to the Espruino. You could then tell it what to do via the usart.
The other option is just to put new firmware on the bluetooth module. I know you can get different firmwares but you'd have to have a look around to see what you can find. I don't know if Adafruit made their own firmware or just flashed someone else's - but it might be that their module uses the same CSR chip as the Espruino ones, in which case you could just flash the firmware that they created straight onto it.
Great answer, thanks! Looks like I'll be buying an EZ-Key anyway, and see what I can get out of it.
Is it at all possible to use a Bluetooth module to pair and act as a keyboard (HID)? Bluetooth's HID profile "is a lightweight wrapper of the human interface device protocol defined for USB" (Wikipedia). As USB HID is already being added to Espruino, maybe I could build on that?
Adafruit's Bluefruit EZ-Key is an option, but it would be great to be able to do this with Espruino only.