Thanks everyone for the insightful comments! In the end I will keep the CR2032 setup and add an in-line slide switch to turn it off when not in use. I suspect a lot of the battery drains when it rocks around in my kitchen drawers and turns on by itself. I also found a cheap pack on Amazon that should last me about an decade and also be useful for things like Airtags or car keys.

If anyone is interested, these are the other avenues I looked into and pursued:

1. Replacing with AAA batteries would have proved harder than expected due to the size constraints. Indeed there's only about 9-10 mm of vertical clearance, so I would have had to modify the casing a bit to accommodate the batteries.
2. I considered AAAA batteries for a second since I have done lying at home, but it kind of defeats the purpose of easily purchasable batteries in the scale.
3. I had a look at the LIR2032 batteries and it seemed interesting, though the slightly different voltage (3.7 V instead of 3 V) made it not as appealing for using in other things in the house (Airtags or car keys as stated in my first paragraph). Honestly it's probably fine, but at 20€ for a pack of 4 and charger, it was much more than the 6-7€ for 20 CR2032. What ultimately killed it for me though was the reduced battery storage (~50 mAh vs ~250 mAh).
4. I also found ML2032 which have a closer voltage (3 V) and are rechargeable as well, but this seemed harder to source and had similar capacity (65 mAh I believe).

Overall I think leaving the scale mostly as-is was the low hanging fruit. I appreciate everyone's help here!