Technically, no. You do not have to do that. That is the auditor's OPINION. Ask them to show you in the standard where it says you have to have the company name on all documents. They can't because it's not there. If you can show you are properly controlling your documents, it should be a non-issue.
One of our sister company's went through something similar where the auditor said they must have bulk trucks sealed with a number and name on the tag because in their opinion, that was the only way to ensure product safety, when in fact, there are other ways. No where in the standard did it say they must use seals with numbers & logos.
My company seems to change it's name more often than I would care for. I now refer to it as "the Company" within the text of all our documents. Most procedures have our company logo, but aside from that not all instructions or forms do. If you were written up for this, you would be well within your rights to appeal it, because that requirement doesn't exist in the standard.