If you have “protected” no claim bonus, I agree you prob want to go straight to insurance.
If not, it’s difficult to calculate the financial effect which will stretch over a few years premium wise.
Without “protected” you will drop back two years bonus and if you are not currently on maximum bonus at the moment, the step- back will cost you over a few years until you build back up.
No such thing as a low cost repair these days.
It’s very galling when it is “no fault” but without another party to recover from it’s just rotten luck.
Hope all goes well whatever your final choice.
Oh, and remember you will probably have a reasonably chunky excess on your policy. Mine is £500.
By the way, I am an insurance broker of (too) many years standing.