If it were me, I would make the call on how well I could document it. If I had the proof, it was easy to follow, and difficult to dispute, I would pursue it. If the company is a solid, legit company, when they see the proof, they will follow though and pay you. If they don't, it lets you know to be very careful dealing with them in the future.
However, (Again, if it were me) if there is any chance that it may be my error, not theirs, and it is a company I plan on doing business with in the future. I would absorb the $12 hit and chalk it up as a learning experience.
If nothing else it would inspire me to document everything more carefully in the future.
Just my opinion, for what it's worth.