Fill out the W-9 providing the EIN and Name of the partnership. The customer that paid your partnership for your services needs either an EIN or SSN of the ones they paid money to, for their own tax return. Under no circumstances and with no exceptions should you ever provide a customer with your SSN number. There are no exceptions. They get the name/EIN of your business, or nothing. When it comes to personal privacy, this is not something that is up for discussion.
