When you claim on your insurance, your insurance company doesn't go after their insurance company, it goes after the other party for the full amount. If they pass on the demand to their insurance company, subject to the limits of their policy, it will substitute itself as respondent and charge them an excess regardless of whether they have to pay your insurer under a knock-for-knock agreement.
You cannot control whether your insurer pursues damages as once you make the claim you subrogate your rights, and they have the right to act as you in court.