Did you document the damage on the spot - photos, video, a report with witness signatures?
You're right that a traffic inspector during an official exam is a full-fledged driver operating your car. That means he's responsible for damage caused by his fault. By law it's either his insurance liability or personal liability. The logic is simple: if you yourself hit a pole, you pay; an instructor is the same deal.
The catch is that traffic inspectors sometimes deny this responsibility, claiming "the owner handed over the car." That's not true, but they can push back. So you need evidence: photos of the damage with a date, preferably dashcam video if it was running, witness statements from the test site, a report of the vehicle inspection. If nothing was documented, things get trickier - they'll argue over who's to blame.
Next step: file a written complaint to the instructor and his superior at traffic inspection with a repair estimate. If they refuse - court is your only option. Court practice usually sides with the owner, but only if you have proof of that specific person's fault. As an option, talk to the department head - sometimes they prefer to settle it quietly to avoid trouble.