I just need a little comfort level. Bank A has a list of established customers that affiliate B would like to market its products to. None of these customers are customers of affiliate B.
Can we avoid the opt-out requirements by introducing affiliate B, its contact and product information, in a solicitation made on Bank A's letterhead and from Bank A?

I think you would be OK having Bank A soliciting its own customers. You probably want to make it clear that the products being marketed are not being offered by Bank A.

The rule allows us the exception to opt-out through Bank A introducing Affiliate B in a letter, per above. However, it does not go on to clarify whether or not this opens the door for Affiliate B to go ahead and solicit the customer.

Can Affiliate B then contact the customer? Any thoughts?

I dont' think that Affiliate B can then contact the customer. I think it's up to the customer that might be interested in what Affiliate B is offering to contact Affiliate B. JMO