Not sure why you would by the Renaissance stylus as it's rechargeable and about the same price as the Kobo. It will go flat. The AAAA in the Kobo stylus lasts about 6 months and a spare one has a 5 to 10 year shelf life. They can be bought singly at about €1 or you can get 6 out of certain 6LR23 9V packs (under €3, so 50c each). The spud on the cell in the 9V pack is the -Ve terminal, so goes in the other way to a labelled AAAA cell.
https://www.amazon.co.uk/RENAISSER-S.../dp/B0987J23K1
vs
100% compatible actual Kobo
https://www.amazon.co.uk/dp/B09HSPQFS9
The rechargeable cell will have a fraction of run time and self discharge.
Similarly, I never use rechargeable cells in smoke alarms, electret microphones, digital test meters (DVM/DMM) and other low consumption or intermittent use gadgets. Alkaline cells. Silver Oxide better for buttons, though larger calculator ones can be Alkaline and single use Lithium for coin cells (about twice the volts per cell anyway).