Is this their exact words? Because to me, 'at the side of the box' does not mean inside the box beside the bag. I agree though that it's strange how they can be so certain that it was definitely put inside the box by video recording. Just checking that you've turned the packaging inside out and can confirm this with customer service to be absolutely adamant that you checked everywhere?
It is a shame because for such a big brand company, why would a customer be trying to claim an extra strap just because when they've already paid for the bag? Like
@papertiger said, they have enough revenue that sending a replacement strap wouldn't put them out of pocket. As an example, I lost my favourite pair of Dior Cannage sunglasses at the Porsche museum in Stuttgart, it must have accidentally fallen off without me realising and even after retracing my steps around the entire museum I couldn't find it again. I think another visitor found them and took them, but I called Stuttgart customer service on my UK mobile number (I get free calls and data within the EU) and gave them my number to call back, although I apologised for them having to call me back long distance once they had checked with the cleaners and reception staff. The customer service agent told me, 'trust me, the international call won't be an issue'.
I agree though, return to the credit card company and explain the situation, that it's not as described because it's a long distance purchase and you've got a certain level of guarantee with items not purchased in person. Also go back to customer service and insist on seeing the video recording yourself after checking the box and turning it upside down to explain that you've checked all contents of the parcel thoroughly and that the strap definitely hasn't been included so you'd like to see the evidence that proves the strap was included in your parcel. Otherwise they are selling you an item not as described and you are due a full refund. In those instances, them sending you a replacement strap will cost less.
It is definitely separate, all online orders are handled via the warehouse in France.
I've already got this habit ingrained in me from living in Hong Kong for many years! Even when I'm told it's a brand new item from their stock in the back room, I still check it thoroughly, but I'm lucky that even though I don't have a Dior store near me in Edinburgh, my SA in London checks everything for me before packaging it and putting it through. He managed to find a massive hole in a Cruise 2022 skirt I wanted in time when he was getting it ready to be packaged before I paid for it! He even put one of my orders through the French warehouse because the hardware on the bag did not meet his standards so I could get a better one