Can you keep the settlement money from the at-fault driver's insurance company and get your company to fix your vehicle?

No. You have the choice of either accepting payment from the other party's insurer, or making a claim against your own collision coverage. If you own insurer pays your claim, it inherits your right of action against the other party through a process called "subrogation". Your auto policy probably states that you would breach your duties under the insurance policy were you to hinder its subrogation rights.