This really isn't a CIP issue as far as the regulation is concerned. The regulation does not apply to a mere signatory on an entity's account whether the entity is publicly traded or not.
What you are actually trying to figure out is the wording in your CIP and whether you want to change it. Publicly traded companies and government entities are entitled to refuse to provide the information, but no more so than a closely held company or a bowling league. If they want to refuse, they can refuse. Any legal requirement applies to your bank, not the customer.
Your bank is entitled to refuse to open the account if you do not get the information you require whether you base your requirements on the law or your own opinion. Some do.
If your bank thinks getting the identifying information is necessary signatories for closely held corporations and LLCs, then it will struggle to figure out what difference it makes that the entity is publicly traded.
In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.