It strikes me that your job is probably fully computerised in larger shops and errors reduced to less than 0.1% say, caused by people putting the wrong things in parts bins, so yeah, that’s your competition – a machine! Also an error on a bag of spoke nipples is one thing but an error on a frame might be more serious.
From an employers perspective, the cost of the error is not just the difference in the cost of the parts; the cost of sorting it out will dwarf that error, especially in small items.
However, it may be that you are annoying your boss in other, less quantifiable ways but this is all he can actually pin on you. Doing much unpaid overtime? Are those around you? That’s the classic one: just not working hard enough.
As for the invoicing: what, to whom, how much, unique ID, group them into sets of similar items if you can, or do in batches of 5 or 10. Swap batches with a colleague for mutual checking.
Your boss probably can do them a lot faster and better. That’s why he’s the boss. And it’s his money you are spunking.