You have a few options within Apple's eco system.
The first is Joint Venture, a program that gives priority service to business customers both over the phone and in store. With Mac repairs this also allows for a loaner if needed for the duration of the repair. You can read more about Joint Venture at Apple's business page. It IS a paid, subscription, service but depending on your needs it could be helpful.
The second would be to look into their iOS Direct Service Program. Essentially entering into the program allows you to train your IT staff to properly diagnose iOS devices in accordance with Apple's guidelines and setup service on them directly, without dealing with AppleCare/Genius Bar. It's a free service but your organization will need to set up normal credit/billing info with Apple and jump through an application process to enroll.
The third option is creating a mail in repair with AppleCare. They can provide a quote over the phone and send you a prepaid box to ship the phone in to them. Shipping is overnight each way (one night to get the box to you, one night for the phone to reach them, another night to come back to you) and included at no additional charge. This is true for warranty and non warranty repairs.
With all of that said... I typically take our damaged devices into the Genius Bar by scheduling online ahead of time. An appointment IS required, however if you show up without an appointment they can often squeeze you in within an hour. They'll provide a free quote and if the part is in stock offer to repair it at that time if you agree to the cost. If nothing else it gets you out of the office for an hour! If I were in your shoes I would do this. By checking the "business related" box in the reservation system you should end up with a member of the business team coming out to tell you about Joint Venture. If they don't you can always ask the Genius or Concierge about meeting with a member of the Business team to discuss exactly how Joint Venture could help in situations like this.