I finished the CR2032 node finally, and put it on the balcony to give it a try. I faced a few problems getting it run, but so far so good, and we'll see in a few days.
It turned out that even a fresh new CR2032 cell does not like high current draw. Those 15mA made it to drop the voltage below 2.7V, and the BOD in Arduino kicked in, resetting the whole thing. I added a small cap (33uF) just to cover any potential current surges, and reduced the current drawn during power up and regular operation (powering down the RF24 radio all the time), and that helped. I experimentally set the start-up fuses to "6CK/14CK + 0ms", and BOD to 1.8V, and we'll see how that will perform.
The picture shows the node reporting the temperature and other info about itself. This LCD is connected to Raspberry Pi and I am using it as a temporary control node before I make a final decision if the central node will run on RasPi, Arduino Due or Arduino Mega. My original idea would require RasPi, but if I drop some requirements that probably do not make much sense, I can easily run it on something smaller. The requirements included checking emails and making the info about unread email messages available to all nodes that care about that, and that would need SSL which is tough on Arduino, but that might not really be necessary.
What I don't like on RasPi is just two SPI devices support... I might get over it by using bit banging, which could work for some situations, or perhaps by adding Arduino Pro Mini over I2C to handle all the RF24 and RFM12 stuff.