If the buyer is being stubbornly insistent, it really comes down to a matter of personal choice with how you deal with it. Personally, I think there are valid arguments for being done with it in the simplest, most time-effective and stress-less way, and also valid arguments for fighting to the end.
I had one buyer tell me they got one less of something than they were supposed to because they had extra of a matching piece that goes with it - I tried to tell them that I always include extra pieces of that, because they're really small and easy to lose so I add them out of courtesy, but in the end I just refunded them $1 to be done with it - about 2 x what the "missing" bit was actually worth... The case that prompted me to keep even more thorough records than I was already sounds very similar to your buyer, too.
I knew beyond a shadow of a doubt that I had sent all items they ordered, but they refused / ignored all of my attempts to resolve the matter amicably and just kept on insisting I send more or give them a 50% refund. In the end, I guess since I wouldn't cave without trying to get some proper details from them, they went straight to PayPal and opened a dispute. At that stage I was no longer willing to make attempts to resolve and just selected the 'return for refund option'.
The buyer decided to be clever and sent something small in a registered letter. I knew what it was as soon as I received it, and that it didn't contain my items, so I didn't open it, instead I called PayPal and requested their advice. Well... They told me unless I could tell them what was inside the envelope, they'd just find in favour of the buyer 
(I wanted a credible third-party witness to me opening it, but the PP rep basically made it seem like I had to tell him right then and there or else). So, the rep listened to me open the envelope (and photograph every stage). Then I sent everything through to them.
As soon as they got all my evidence, they immediately found in favour of the buyer but said that due to "unusual circumstances", they wouldn't take the funds from me. So, buyer did get a refund, but I didn't pay for it (small comfort, they didn't deserve anything, though I reported them to ACORN at least 
).
So, for me, it's the pre-emptive methods that are the most effective, and sending through the photos to show exactly what was packaged has been the most effective (it was pointless with the $1-refund buyer I mentioned above, but for everyone else since I started doing that, they just haven't been able to come back with anything....so far *touch wood* 
).
