With respect to your BRC/IoP certification, the auditor's position is likely to be correct.
If your labels are not for direct food (or other high risk product) contact then you would have been Category 2 or 3 for Issue 3 of the Standard. Only Category 1 had the requirement for scalp or beard/moustache covering, and then only if a hazard analysis had identified a risk.
If you do make labels for direct contact, you will fall under the High Hygiene risk category of Issue 4 of the Standard and again you only need to wear them if a risk has been identified. (6.56 & 6.57)
If you don't make for direct contact, Issue 4 has made it a bit more complex in determining which risk category you fall into, partly because of the BRC's inconsistent use of the words "absolute barrier" (in the decision tree) and "impermeable" (as an example in Appendix 2)
For example, if you made self adhesive labels which are applied to a paper or perforated film bag used for bread such as in an in-store bakery you would come into the High Hygiene risk category.
Your new owners may have different reasons for making you wear hair covering outside of the BRC requirements - standardisation, marketing etc (or even just lack of understanding) and in reality, if they make the rules you have to obey them.
The supermarkets universally take the approach of do as I say, not do as I do, and some of the practices I see in stores makes me cringe. Washing trollies and baskets is just one of a long list of things that would not be permitted by the BRC or even the supermarkets' own technologists if they saw them in a supplier's factory. Unfortunately, they rule us.