A company previously had one application that customers could use to access their account, pay their bill etc. Alternatively, customers could call, and an employee would answer the phone and do all of that work for you. That employee had a separate interface, an enterprise app for expert users. Recently, the company integrated these systems, so now internal employees are using the same interface as external customers. As a result, the time that it took for employees to complete tasks went way up. They've asked if there is research into the pros and cons of using different interfaces for internal and external audiences completing the same tasks. Has anyone seen any studies about this?
Companies frequently make this mistake for naive reasons. Examples: "our internal users should understand what external users see", "we should eat our own dogfood", "it's a cost saving", "it would betray our brand if we use other software".
The truth is: the difference between internal and external users is the same as the difference between any two user groups.
Therefore, UX suitability follows the same principles as with any other user groups. If the workflow and needs of your internal guys are similar to those of your customers, then it's more likely that the same UX can be used. But, in the real world that is almost never the case. Usually internal users have expert capabilities and/or different roles (e.g. admin, account management) so external UX is rarely appropriate for internal use.
Arguments around brand fidelity, product accountability, etc. are usually based on a poor understanding of customer service (e.g. Ferrari employees don't all drive Ferraris, and customers wouldn't benefit if they did), and an obtuse desire to place corporate needs above customer needs (if a different internal UX will serve the customer better, then that is the correct solution, period).
You asked for specific studies, and this is a long way of explaining why I don't think those studies exist in the way you think they do. Once you accept that internal vs external is no different from any two user groups, then it follows that what matters more is the difference in workflows so there is no reasonable way to do a study of "internal" vs "external" because the use cases vary dramatically between companies.
Any comparative usability study or approach can be used to measure the differences for your company's specific flows.