My wife's minivan was rear ended in April this year, fixed a couple of weeks later, and then rear ended again in May. Both happened at intersections at red lights, hit at lower speeds by inattentive drivers who were not paying attention. I have a front dash cam, but not rear... yet. Both times I went through the other party's insurance and not mine. First was Geico, second was Bristol West. Took the van to my choice body shop the day of each incident and called the other party's insurance on the way there to open a claim. Both companies called the insured to validate their fault right after I opened the claim. Geico got me a direct billed rental car, Bristol West did reimbursement. Both insurance companies paid the body shop directly. I barely had to do anything to make sure they took care of business. BTW, I'm in Florida.
Now, having said all of that. If you have injuries you need to get that looked at ASAP. You have like 10-14 days to report injuries, may be different in your state. In Florida your insurance covers your own injuries, with the other party's covering when you go over an amount, regardless of fault. Get familiar with your state.
Since you were hit by a company vehicle, I'd go with the lawyer advice as well since that company will try to get out of their insurance premium going up. Key is getting the owner to admit fault so their insurance starts to cover things. While my wife was hit by little old ladies who had no issues accepting their personal responsibility, companies will make you prove things every step of the way. Was there no police report that spelled out what happened and who was at fault? Was the other driver ticketed? When I talked to the other party's insurance company I gave them the police report numbers and they took it from there.