Keep track of open trouble tickets with the carrier. Ask that your central support site be kept informed of problems. Obtain any documentation, product orientation, or training that the ISP or carrier can provide to gain insight into procedures or the equipment they use. Retain phone numbers and contacts of network support centers and technicians, and never pass up the opportunity for an inside number. Knowledge of the circuit and equipment will make troubleshooting a team sport, not a finger-pointing exercise.
Facts have a way of overruling bickering when used constructively.
Find a support technician at your ISP that you can contact directly. This helps prevent being trapped in a caller queue for hours. Also, contacting the same tech builds a relationship with that person. It's likely they will take a personal interest in getting your problem resolved. If the survival of your business depends on the carrier or ISP, you need to meet the people who work there.