I helped build a similar one one that traveled around my friend group for a bit and now does 3 days a week at the local brewery. Ours stays under 150°F... After a 10 hour day of continuous use in 90°temps.

the exterior is always comfortable to the touch- some times i climb up on top of it and sit and watch the crowd/ music/ call out orders for the guys if its busy.

They're usually that layer of bricks, a layer of insulation, then like 6-8" of cement. The whole point is to keep the heat reflecting on the inside.

If the one you have access to is reaching more than 200°F, it means the interior has cracked somewhere, and it shouldn't be used until repaired. Your risking the insulation catching, or burned through it already, contaminating the pizzas, and rhe exterior concrete layer being damaged.