Open an Account in the Cayman Islands
The Cayman Islands have a similar banking system as that of the USA. Their secrecy is also well known. As a result, your corporate name is the only information that the bank will require. If you are worried about your privacy, there are ways to ensure it. Create a series of corporate names and trace them via a holding company that does not lead to you. Although banks from this jurisdiction do not encourage evading tax, they also do not inform about deposits on the account. This provides customers with a sense of security. However, in cases of national emergency or pressure from the US government, there have been several incidents when it has allowed access to help the government. In this case, your privacy is not guaranteed.
Contact banks in the Cayman Islands and discuss your requirements in confidence.