I went through the motions and saw that you get redirected to update the payment and pick-up address for the purposes of the invoice, which doesn't actually exist in eBay's list of addresses on file anymore, so I'm guessing that's the cause of the error and that the address that shows on packing slips is now tied to a different address on file.
I went in and changed both my return address and dispatch-from address to have my shop name in place of my personal name, then tested what shows up when I went to print a packing slip for a current order, but it still displayed my personal name.
Once I get a new order, I'll check again just to make sure it's not because that was the address I had set up when the purchase was made, otherwise it'd have to be tied to registered address (i.e. the primary address used for account registration, not necessarily the same as the one you enter for dispatch, returns, or deliveries - I didn't change my delivery address as I thought the chances of it being tied to that are low, but typing this out I now realise I'm talking about eBay, so it's probably a 50 / 50 shot between that and the registered address 
).
I wonder it it comes from when they did have payment addresses as an option, and now can't be changed at all because it's disappeared from the addresses section on ebay? (The removal was relatively recent, I'd say within the last year, anyway, unless I just haven't been paying attention - entirely possible 
).
I do remember adjusting all but my registered address to my PO box a long time ago, so that any details given to or requested by buyers wouldn't result in them obtaining my home address, but I've always entered my full name so there weren't any differences between each one.
I'm starting to think it's not the registration address, because I realised that when I went to the edit screen from the invoice screen, it was displaying my old mobile phone number, which was updated on my registered address over a year ago. It may just be one of those eBay mysteries 
