Because I was on a deadline, I chose to reset the reader after a certain number of readings (can’t remember if I opted for 19 or went with 10, just to be on the safe side). That seemed to have solved my problem (although the exhibit it was meant for hasn’t been opened for the general public jet, so fingers crossed).
Out of curiosity I did try to flash the reader with an altered sketch. I got there in the end, but it requires some fine soldering, because one of the pins isn’t broken out. If you can manage that and know how to flash the Attiny using an Arduino, you should be able to do it too.
But shame on Sparkfun for not recognizing this flaw and instead telling people to “just edit the firmware” (Yes, I know this stuff is for hobbyists. But isn’t it a bit strange to expect someone to go and solder jumper wires to programming pads on a product that was designed specifically to be more user friendly and make the hobby more acessible, like the Qwiic range of products?) or just go out and buy a product that’s more than 10 times as expensive (mind you, I was using 5 readers, so “just buying the better product”, would have meant investing more than $1500,- in readers.)