the SCL pin of the K24C02C doesn't seem to be connected to anything else on the PCB. Is that normal?
No it is not normal nor is it the case here.
The trace goes under the IC and emerges on the bottom side in the middle.
The IC seems to still be connected to the chip in your scales. You need to disconnect this chip (cut the traces ) before you can hope to have something else talk to it. You should also put a 4K7 pull up to 5V on both lines and remember to connect the grounds together.
Excuse me - I'm fairly new to this. What do you mean with "The IC seems to still be connected to the chip in your scales." What chip/traces do you mean?
I want to keep the scale working and only "sniff" the live measurement data from the scale. Won't cutting traces mess that up?
Could you elaborate this a bit? Thank you!