I have no experience with this but worked with a guy who did.

His contractor came back a month later wanting $1000 for work he forgot to bill for. He emailed and talked to the guy and demanded that the contractor show up, do a walk though of the entire job and outline every item of work that was done and give him copies of receipts for everything that was bought for the job as well as a responsible explanation of where it was used. The contractor baked and threatened to put a lien on the house. The owner said that was OK but he was going to have to come through with most of the demands on a walk through and a write up of ALL the work that was done BEFORE he got paid.

After a lot of back and forth they negotiated down to $300 AND a lien waver from the contractor. So he paid him off.

This is likely a losing battle for you. It will cost more to fight it then it will to pay for it. But it doesn't mean you can't go down without a fight. At very least he needs to provide a lien waver to prevent him from trying to extort any more money from you.

A LOT of contractors are terrible bookkeepers and this may indeed be a genuine reflection of work. Or the guy may just be trying to put the squeeze on you for more $$$. It's impossible to say. You do need to protect yourself and make sure he understands you are not going to just bend over and take it. After you pay him and get the lien waiver - which also needs to be gotten from an subcontractors he used as well - then turn to social media and the BBB. Just make your points known and express your experience succinctly. It may prove for naught, but it may save someone else from the experiencing the same thing from him.