Is your insurance zero dep or comprehensive? If it's zero dep then all you will have to pay is 1000-1500 bucks which are filing charges and if it's comprehensive then you will have to pay around 40-45% of the total repair cost. Even I also had 'Anjali' written on the front right door of my Laura and the scratching was really deep and in a poor handwriting
; I went to A.S.S. and had a word with SA who simply told me that I shouldn't apply for the claim; half an hour later I returned back with a badly dented door and quarter panel, all I had to pay were filing charges, rest the insurance took care.
In case you file an FIR then the repair will take time I must tell you. Your insurer will tell you that they will process the claim only after the local police submits their report after their investigation. Now the local cops will take their own sweet time in giving the report that they couldn't find anyone and after that only your claim will be processed and refunded.
In case cops find the culprit, then the settlement time will get even longer because your insurer will want the culprit to pay for the losses.
I hope you are aware of these sides also of filing and FIR and then going on for insurance claim.
I have some other ways of getting the work done via insurance but I doubt if you can really do that to your brand new car.