Under the "improving batteries improve Electron" thesis, where installing improved batteries functionally improves payload with a small amount of effort and minimal rocket redesign...
For the original 220kg payload Electron, stage 1 supposedly is a 1MW powerpack around 200kg, but I don't have a handy energy density number for the current pack. Somebody's rough guess estimates were
F1 engine 6770kN thrust and 41MW turbo-pump = ~6kW/kN pump power per unit thrust.
For the 1st stage of the Electron that translates to 6kW/kN x 183kN = 1098kW
With ~160 sec burn time from full duration engine test that requires 1098kW x 160/3600hr = 49kWh.
Lithium Polymer batteries have energy densities up to 265Wh/kg = 185kg battery.
which mostly squares up with the 200kg stage 1 battery pack comments.
but someone else pointed out each rutherford is 2x50hp motors, so power is 671kW, thus 29.8 kWh needed?
Stage two burns for 5 minutes with one engine (183kN/9 so 122kW), so 10.16kWhr. Original 265Wh/kg guess leads to a second stage battery pack mass of 38.3kg. Some comments have said overall battery pack mass is 250kg, so with S1 at 200kg you can guess S2 is 50kg as a rough approximation, which roughly fits.
So, Amprius is announcing 500 Wh/kg, 1300 Wh/L per cell type lithium-ion batteries
https://amprius.com/the-all-new-amprius-500-wh-kg-battery-platform-is-here/So while that is per cell, which gets worse as you build up a whole battery pack, but for the sake of numbers that's roughly double the power density. But for a first stage, you get roughly a 10kg mass drop to 1kg payload gain. So roughly 10kg of payload gained from stage 1 improvement. But second stage is more 1:1 ratio.
So rough guess, if original 220kg payload Electron were getting these new batteries results in at least (100kg/10 and 25kg=35kg) payload boost WITH NO OTHER IMPROVEMENTS. 16% payload boost.
But Electron got batteries upgraded at least once (when 400WHr/kg existed I think), which along with other improvements, had increased payload to 300kg (80kg payload improvement).