They must be referring to the consignment note, certainly not ideal given that you are trying to protect your source. What type of products are we speaking about, is it something you would expect to get repeat orders on?
If you anticipate them being one off sales, then having the supplier's name on the note probably isn't a really big deal.
Not sure if they would do it or not, but might be worth asking if they would use your details for the consignment note. Mind you, the courier service may well have an issue with that, but won't hurt to ask and see what they say.