This guy understands insurance.By insurance law this falls under permissive use. If you and hand your keys to your brother/friend whatever to take your car to the store you have given him/her permission to drive the vehicle. If he gets into an accident on the way to the store your insurance will cover damages or the other parties insurance if he was not at fault. His insurance is not responsible. Now if he grabbed your keys without your consent and left it is on him so long as you are willing to make a statement that he did not have permission. (Could be other legal ramifications for him with this).
As a general Rule most insurance companies will tell you to always FILE A CLAIM WITH THEM even if you have the other persons info and they appear at fault. Fault can be 100/0, 0/100 (rarer than you think), 50/50, 70/30 Insurance companies may not have your best interest in mind but they sure as hell have theirs and have a team of trained insurance law adjusters and lawyers who will go to bat to ensure the other party pays for damages if liable.
If this shop is insured and the loss type is covered I believe that would apply over permissive use. Personally If it were me I would have filed a claim with my insurance right away, payed the deductible and got my car fixed or replaced and let them reimburse me the deductible and any other $ once they go after Titan's insurance and get reimbursed. It could easily take months where your own company should take weeks if that.
Your own company is extremely unlikely to try to deny the claim. They are going to pay it and then try to get their money and your deductible back from anyone they can possibly blame.