We have different vetting standards for our supplier approval program when it comes to customer-prescribed ingredients - overall our customers do not provide us with transparency into their supply chain (and frankly we don't want the headaches of managing their supply chain either) and so we don't ask too much about that. We primarily provide the service of cleaning product and co-packing according to our customer's request/specifications.
Yes, we print labels on-site - the customer provides us with a PDF file of their label and we print them on blank label sheets.
What you just expressed is exactly the issue that come across... we do have our customers sign an indemnity release that states that its ultimately up to the customers to ensure their labels are accurate. However, our QC staff runs the labels through a fine tooth comb and any discrepancy between the new label and the previously approved label provided by the customer is noted.
It seems to me that you are doing all the right things. If your customer isn't supplying you with transparency, then you have to rely on the information they give to you. If you are doing any additional blending or have processes which would alter any of the information on the label, it would be your responsibility to make sure that the label is accurate to meet their processes and yours. For example, if you produce product on an allergen containing line and you choose to offer a disclaimer per company policy, then you need to be sure to add this disclaimer to the label and get customer approval as well.
The fact that you are vetting the labels tells me that you are also further following processes by ensuring accurate information. If you don't already have the process in writing as you have described it here, I would suggest you do that. I feel that your checking the labels is proof that you are confirming as much about the label as you can. Obviously if there is processing upstream of you, and they aren't transparent about those processes, you can't know all there is to know about the information on the label. However, if you do get wind of regulatory changes that could affect your products, you are responsible to make sure your customer is aware of these changes and to make sure the labeling meets the proper standards.
This is one of the clauses in BRC that annoy me, because this really only makes sense for people who totally manufacture their own product and provide private-label product to other customers. This doesn't really make sense for manufacturers that co-pack for other companies, and don't own the formulas. If you are provided with a pre-mix of proprietary (%) ingredients, you can't know all there is to know about the labeling. You can do your best to provide accurate information from your processing point only.