Of course the lessee is obligated to indemnify the lessor for any damage TO the rental, whether it resulted from A’s negligence or B’s negligence. And, contingent upon the language in the lease - which you alone seem to be privy to – the bailee might be classed as an insurer of the condition of the vehicle.And as already pointed out, there's the fact that the renter is contractually obligated to assume responsibility for anything that happens TO or WITH the rental car.
But where does this “WITH” business come from? Where is it “pointed out”?
Are you now claiming that because of the rental agreement A is “contractually obligated” to account for property loss suffered by the owner of the other vehicle? That the other owner has a cause of action against A for property loss upon a theory of contract law?
What an absurdity that would be. As is your suggestion that any language in the two-party rental agreement could alone render A liable to the owner of the other vehicle.
Noticeably missing here are your authorities sustaining your “theory” of “negligent entrustment”! Or that operating a motor vehicle without a valid permit is negligence per se and negligence imputed to A due to permissive use.
Each of which theory you injected to support your contention of vicarious liability on the part of A. And again without statutory or case law supporting authorities.
But do keep up with the effort, please. Because the more you do the more apparent is your shortage of knowledge of the subject matter.