[werk_ag]

And / or a significant fraction of the power use is caused by running the PCB on board regulator at 12V instead of 9V (I saw a heatsink on one of the PCB's, can't remember if it was the TX or RX or both). That's my guess.

If you are concerned about the power loss at the 7809 regulator (less than 0.2W when the fan is not used), you can bypass it and feed the board with a 9V power supply.
As uncle bob said, the heatsink is only needed if you use a fan in your Radiation Shield. Even if you install a fan, it don't have to be always on. Software allows you to define on/off conditions.
According to this study, a fan in a RS is only beneficial in some circunstances.

Powered with 9V the TX module draws around 30mA, it's a lot of power I know, this is why the RF link is so reliable. With a good antenna it's almost 10mW TX power.

Now I'm using a 14Ah sealed lead acid battery designed for solar systems (I don't remember brand, it's the grey one in second photo), mainly because I don't want to be worry about long winter nights or several days without sun!
Solar Charge Controller also draws some power. So the battery capacity is mainly defined by the number of days you want your system runs, when the battery delivers more power than it receives.

A solar system is not required, the system can simply be powered by a 220Vac to 12Vdc power adapter. Total consumption (without fan) is around 100mA - 110mA