Who on earth reads family court decisions, except the people directly involved? I don't think this is about public perception, it's about controlling the narrative. If the alcohol testing isn't in the written document, then there's nothing that contradicts his narrative of "I'm a great dad being persecuted by my controlling ex".
I know we're not supposed to go heavy on advice on this forum - but I suggest pushing on this point (including testing in the parenting order). If the testing is reciprocal (you will agree to test too if he asks) and if costs are assumed by the person who requested the test in the event of a negative test, there is no downside for either of you to agree. If he really isn't drinking when he's with his kid, the tests offer him an objective way to prove it at no cost to himself.