This used to happen to me all the darn time when I started selling craft supplies, and I couldn't always remember beyond a shadow of a doubt that I had packed the order correctly, so I would just post out whatever the buyer claimed was missing. 😓

But the frequency started to get really suspicious to me, so I decided to take photos of every single order I send, with the addressed package and all items that were about to be packed into it. This was the only way I could think of that gave me clear evidence I could refer to to confirm a buyer's claims. Now, if if I get a message like that, I look up the photo - sometimes the item they say is missing is not in the pic, so I don't have to doubt what they are saying or feel suspicious of someone etc, I can just apologise and tell them what their options are (and I can tell it's a relief for the genuine buyers, as they are often worried they won't be believed). 

If my photo shows everything was there, I re-check my packing area to make sure nothing slipped down somewhere and went overlooked - a couple of times I've found stuff lodged between boxes etc. If it doesn't turn up there, I ask them to double-check all of the packaging and so on, since my records indicate everything was included - I also send them the photo so that they can be assured of this and not just dismiss it out of hand if something really was overlooked when unpacking and they genuinely believe it's missing from the order. About 7/10 times I do not get a reply to that, the rest of the time the buyer just lets me know they found it. 

It is tedious and time-consuming at high volume, but it's just part of my routine now and has saved me a lot of time in other ways, plus quite a bit of money and unnecessary worry over buyers who were telling the truth. 

I know this might not help directly with the current situation, but hopefully there's something useful in all that up there - and for now, if you are very confident everything was included, you need to be able to demonstrate that to the buyer without being accusatory - sometimes (if they are not being genuine) they are just looking for a quick "I'm sorry, have a refund" response, and if they don't get it, won't bother pushing back.