I have few thoerys of my own why it was not done that way.
I could think few thoeries while possible that the Dominion could pick of a starship doing it. The Dominion send ambassdor asking starfleet remove the minefield before start a attack suggestion that if UFP could done lot sneaker they might have even aviod Deep Space Nine been attack first place seem put that thoery to rest.
UFP over careful with Treaty of Algeron which forbid starfleet vessil having cloak device put on it. We did not know details of the Treaty of Algeron. It could been away were limit what USS Defaint be allow us the cloak device for lay mine field off limit Romulus Star Empire was unwelling bend fearing a war or tech issue Federation never design mines that safe cloak and total active as protective means to aviod be excuss violence the Treaty of Algeron. While this possible is back good amount evidence the fact Klingon did having cargo ship Deep Space Nine Sons and Daugher of the Empire equement cloak device suggestion it was beyound the Klingon techology just rep