I am using a Yuasa 44B19L battery:
It is almost half the weight of the OEM battery and still useful in a daily driven car. It fits perfectly in the BRZ RA's battery holder (Japan spec). So, no need for custom solutions.
I replaced the following Subaru parts. Part numbers are for the new ones:
Tray-Battery
82122CA010 x 1
Rod-Battery Fix
82161AG001 x 1
Holder-Battery Fix
82182AG010 x 1
Dimensions of the battery:
Length (mm) 187
Width (mm) 127
Height (mm) 225
Any other battery with similar dimensions will work.