Sedgwick are a claims management company. They won't take it on until the driver has admitted liability as they'll be out of pocket otherwise.
I am confused why Enterprise are screwing over their insurance company who will have to pay inflated costs from Sedgwick.
I would want to be in touch with Enterprise's insurers (unless of course it is them who outsources claims handling to Sedgwick).
edit:
Reading more about them, the latter is most likely true.
https://www.sedgwick.com/solutions/global/uk
An insurance company who actually outsources everything are using Sedgwick.