A single customer can only be assigned to one AWS Support Partner Billing Rule at a time. If you cannot see the customer in question listed under the dropdown, please check that the customer is not already assigned to an existing AWS Support Partner Billing Rule.
Another consideration is that an AWS Support Partner Billing Rule can only be applied to a customer who is being passed through the Partner Generated Billing engine.
More info regarding AWS Partner Billing Rules: Using the Partner Platform as AWS Service Provider