Deciding if you want the insurance claim for your totaled vehicle to go through your insurance company or the at-fault person's is a personal choice. It is usually better to go through the at-fault party's insurance carrier if you can though, especially if they have already started working on the claims process for the incident by reimbursing you for your medical expenses and rental car costs.

If you go through your own collision coverage you will have a deductible to pay as well as this incident listed on your claims history with your insurance company. If your company does pay on the totaled vehicle they likely would then subrogate with the at-fault party's insurance company for the expenses they paid out for the settlement of the car, towing expenses and perhaps try to get back your deductible amount as well.

If instead you continue the claim for the vehicle with the at-fault party's company you do not have the deductible to pay and the claim would be on the other party's history since their insurance company would be paying out for the settlement.

Since it appears you already started a claim for the car with your insurance provider and other claims (medical and rental car reimbursement costs) with the at-fault party's insurance, I would suggest contacting your company's insurance adjuster to see if your collision claim could be stopped so that the claim for the vehicle would go through the at-fault party's property damage coverage instead.