Sony batteries smackdown Canon batteries when it comes to mirrorless but Canon BP batteries that they use in their cinema cameras last forever.
I think Canon made a mistake by making their mirrorless batteries backward compatible since DSLRs require so little energy.
I see no reason that Canon can't come up with a competitive mirrorless battery but I do not think it can be an LP-N type.
"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 but 90Whr due to 10.8v so 95% more capacity than the R5. Canon didn't update the LP-E19 battery since 2016 when it was released in 2016. The 1DXiii didn't really need more capacity
If Canon reuses the same battery in the R3, it could increase the density similar to the +15% in LP-E6NH or could be a new battery as the R3 is physically smaller.
Are BP batteries better LP-N type batteries?
LP-E6NH is 38.4 x 21 x 56.8 mm. Not exactly rectangular in shape but approximately 45cm3 in volume @7.2V giving a power density of 340mWh/cm3
BP-A60 is 41.5 x 82.5 x 69.7 mm and approximately 240cm3 (5x the LP-E6 size) with 6700mAhr @14.4V giving a power density of 402mWh/cm3
=> BP-A60 is about 18% power density than LP-E6NH but BP-A60 is "much more rectangular" so perhaps that accounts for the difference in my rough calculations