My employer, "company A", has me on an unjust PIP. I am applying for other jobs. A customer of theirs has an open position which is similar to mine. I have done work for that customer (while at Company A and a previous company), and I have visited them onsite as an employee of Company A. The PIP is not related to any work I did for that customer.
Company A does not let me disclose the names of their customers to anyone. I think this is true even of customers who are listed on their public-facing website. The customer is a large company. The open job is in a different department from the one I visited.
May I apply to the job with the customer before I am let go? After? And if I do, can my cover letter mention the fact that I've already worked with them?