The best I can determine is that the three digit number is the total output capacity in milliamp-hours (mAh). At least, I found 2032 batteries with a 135 mAh rating. However, I was not able to find any with a 122 mAh rating. Frankly these are so close that I would be surprised if a device that functions on one would not work on the other. Especially in your case, as the 135 mAh version would just have more total juice so it would last longer. Not a different voltage or anything. Also, I see 2032s rated at 224, 240 and 190 mAh from Duracell and 190 mAh from Maxell. All of them are 2032s.

Does the manual specifically say NOT to use the 135? Or just that it needs a 122? If the latter, I'd use the 135, it should be fine.

Those numbers on CR2032 batteries typically refer to their dimensions – like diameter and thickness. Your electronic scale might be designed specifically for the 122 variant, which is a bit smaller than the 135.