"smackdown" isn't a great scientific term. The A1's NP-FZ100 battery has 2300 mAh and the R5's LP-E6NH has 2130 mAh so about 8% difference. They are practically the same size as well.
Power demand would indicate that the usage of the available power is the key difference in CIPA etc specs between the 2. The A1 processing power efficiency and the combined SD/CFe type A slots are the only other significant difference in usage.
The 1DXiii uses LP-E19 with 2700mAh (10.8v) so only 27% more capacity than the R5 although it uses same space as a vertical grip with 2 batteries. The 1DXiii didn't really need more capacity

The R3 is physically smaller than the 1DXiii so its new battery will probably be smaller that LP-E19 but hopefully higher density ie by using the available integrated grip space efficiently
Why would BP batteries be used in the R3?
LP-E6NH is 38.4 x 21 x 56.8 mm. Not exactly rectangular in shape but approximately 45cm3 in volume giving a power density of 47mAh/cm3
BP-A60 is 41.5 x 82.5 x 69.7 mm and approximately 240cm3 (5x the LP-E6 size) and 6200mAh giving a power density of 26mAh/cm3
=> LP-E6NH is getting close to 2x the power density of the BP-A60