Thanks for your feedback, Gordon!
The watch case looks nice, so we may give that a try. If so, we'll let you know if we need a spare plastic case.
The DS18B20 TO-92 temperature sensor looks promising.
Turning it off between readings will indeed extend the battery life. The monitoring should only be necessary for some days in a row (and perhaps even only during the night time), so the battery should not be a problem.
We'll discuss the possibilities and let you know the outcome.
Thanks again for your help!