"there will always be some moron that will screw the pooch and create major embarrassment"
And when they do, this new reg will allow an easier prosecution for "Conduct Predjudicial to the Good Order and Discipline of the Armed Forces"!
I don't know about the UK, but the US "Uniform Code of Military Justice" has that as the title of Article 134... also known as the "catch-all charge".
While charges can be brought under 134 for this kind of stuff, it would be subject to legal challenge on Constitutional "Free Speech" grounds. Although there would be little chance of avoiding conviction using that defense, something like your reg would eliminate all possibility of using that defense by providing a specific case of "Disobedience of a Lawful Order".
The problem, as you say, the the difficulty of enforcement (which will be, inevitably, selective... usually going after the worst offenders), and also the long-term implications for society as a whole, because where one governmental branch goes, the others will want to follow.
It is also subject to abuse, where it is used to target individuals who are doing what everyone else in their unit is doing, but have annoyed someone who then turns them in... "The British Video Inquisition" anyone?