You own the car and you are the insured so it's your choice if you claim. But.... if they've towed it away and are paying an assessor and storage then if you say you're not proceeding they may charge you, plus if you hit anything, the insurers of whatever you hit will want paying.
The other point is what status they give, different categories exist, some are unfit for return to the road, in which case they write it off and pay you money. If it's considered repairable then you can buy it back, that way you are paid the cars worth, (according to the insurer), but buy back at scrap value and can then repair it. If you choose that option, DON'T send your log book off till they agree, the other option is that you can tell the insurers of the independant opinion you have and ask them to use the bodyshop you've chosen?
(I wrote off a supra and a skyline GTR - not at the same time - and it can be a pain going through the rigmarole).
Good luck with your choice.