This sucks ...
... a bit too much power. The Bluetooth module is supposed to draw only 12mW, but it drained my (admittedly very worn out) rechargable battery pack in 10 hours. And this while the Newton is off. Frank Gruendel has measured that the Newton draws about 1mA when off, which probably comes to around 5mW. Clearly, something is wrong. But after Reading The Fine Manual - which always seems to be a good idea, if you find it in the packing material -, I noticed that the RX and CTS lines need a 1k resistor. I'm now letting one Newton run down it's battery while switched on, and another one switched off but with the module sans RX and CTS lines installed. This should give me an idea how much energy my battery packs still holds and how much the module will draw.
In the worst case, I'd have to add a switch. Ideally however, this problem (if it still exists after adding the resistors) should be solved by controlling the power to the module via the DTR line. Unfortunately, I don't have neither the tools nor the skill to build something myself - it's already quite astonishing that I got this far!
The final setup I'm planning to use also requires a bit of Newton case modding. The antenna should stick out of the case a bit because the case is shielded from the inside with some sort of metal layer. I scratched some of that off in an area where the module sits, and the range is actually quite good. The antenna is now there where the plastic cover for the SER-001 (or modem or audio-in or ...) was. Not too attractive, but hidden behind the port door all the time. A bit epoxy modelling might take care of this later. For now I'm enjoying the first Newton with a built-in Bluetooth module, surfing the web wirelessly!