An assembly is provided for an aircraft propulsion system. This assembly includes a cowl door movable between a closed position and an open position. The cowl door includes a structural panel and a mount. The structural panel includes an inner skin, an outer skin and a cellular core. The cellular core is connected to and arranged between the inner skin and the outer skin. The mount includes a base and a coupler. The base is connected to and arranged between the inner skin and the outer skin. The coupler projects out from the base.