You don’t need to send the invoice. If you think of a retail outlet for example, they don’t issue any invoices. Option 2 is often used by shops who use QuickFile.
I’d always recommend to have copies of the invoice just in case there are ever any questions raised by HMRC in the future. What I would suggest, is perhaps send the emails between you and your client to QuickFile and attach them to that invoice so there’s some proof that you offered an invoice and it was rejected. You would need to set up the email your receipts option, but this guide should be able to help
If you know their company name, I would add it just for my own record keeping if nothing else. It certainly wouldn’t do any harm to add it.
Hope that helps, but if you have more questions, please let me know 