The question turns out to be a bit misguided as the first stage batteries are not ejected during flight; however, it can still be answered no.
Spaceflight 101's quality can be mediocre at times. I think this article is an example of that.
I found no evidence that states that the first stage batteries are dumped overboard during flight. The Electron Payload User's Guide explicitly states this for the second stage (emphasis added):
High Voltage Batteries (HVBs) batteries provide power to the LOx and kerosene pumps [...] During the second stage burn, two HVBs power the electric pumps until depletion, when a third HVB takes over for the remainder of the second stage burn. Upon depletion, the first two HVBs are jettisoned from Electron to reduce mass and increase performance in flight.
There is no such statement for the first stage. Additionally, looking at the layout of the vehicle in various pictures I see no clear mechanism for ejecting the first stage batteries short of also jettisoning all 9 engines. From the User's Guide the batteries (power pack) are located between the tanks and engines, probably around the engine plumbing:
Close up pictures don't seem to show anything either from the bottom or side:
Credit: Rocket Lab, from Spaceflight 101
Credit: Rocket Lab, from Spaceflight Now article
Credit: Rocket Lab, from Engadget article
The incineration part is basically DOA because we know that Rocket Lab is pursuing recovery and eventual re-use of the Electron booster. They have already fished-out a flown booster and because the batteries remain onboard they are not exposed to the heat of re-entry.
The main concern for Rocket Lab is the heating on the engines and they make no mention of protecting the power pack. I suspect that there is very little heat transferred to the batteries during the brief, but fiery re-entry period. Additionally, the toxic flames of burning batteries could damage electrical connectors (among countless other things) and would also be a risk to humans involved in any recovery operations (and protective instruments are not seen in limited pictures).
This screenshot from Everyday Astronaut's Rocket Lab tour video appears to confirm the radial battery placement and definitely shows just how impregnated they are inside the vehicle:
It also shows the individual "battery units" covered in (presumably) some kind of multi-layer insulation. Batteries have some of the strictest temperature limits of all space hardware (slide 5)*. Such tight limits necessitates ample engineering effort towards their thermal control.
*this table is likely referring to older nickel based space batteries, but lithium based batteries still have tight limits.