Currently, the permission needed to allow customers to choose their own approvers, is to make them able to search "Any customer or organization" as below:

"If customers will choose approvers before you begin, make sure your Customer permissions > Who can customers share request with? setting is on Any customer or organization, by searching in this project. This ensures that customers can search for approvers."

However, this exposes all customers in any organization which is not desired result for many customers.

Suggestion is to add an option to search only in the same organization as the customer.