How much influence does a business insurer have over whether their insured settles? I would guess if it were up to them they are going to be fairly mathematical about it, and weigh the odds, and costs together in determining a fair settlement offer. I would feel the strength of the defendant's case is a factor, but not the primary factor by any means.
It would seem to me the insured is in a terrible position if it is purely up to their insurers as I would imagine even settling a claim can make future renewals of insurance potentially unaffordable. And I would imagine just as with car insurance, that business insurers can share information about claims from specific entities or individuals, making it hard to find another reasonable insurer. Does a defendant have any protection from this?