Can I sue the other party's insurance company for bad faith if they refuse to accept a reasonable settlement offer from me?

No. Only the policyholder can sue the insurance carrier for bad faith, because the covenant of good faith is based on the contractual relationship between the two. If you are involved in a personal injury lawsuit against an individual or a direct action against the negligent defendant's insurance company, you will be compensated for your damages if you can prevail in court and have a jury verdict entered against the defendant.