My insurance would have paid it and then fought them, but you are out
your detectable until they win or settle.

That is why you submit it to your insurance.  You can have it repaired immediately and then will get your money back in less time than it takes you to settle it without them.  Either way, your insurance will know you were involved in an accident.  What is the purpose of buying insurance if you are not going to use the services they provide?  Yes you will have to pay the deductible until they receover, but that is better than paying the entire repair bill until they settle and hoping they cover it all.