Unless something has changed very recently, none of the Service Chiefs gets to decide who does what - they are tasked with generating Force Elements @ Readiness and providing domain advice to the supported commander, in this case COM STRATCOM. CJO and PJHQ work out which FE they require based on all the factors in play and the CofC is then tasked to provide units accordingly. Using a perceived failure to deploy the RAF Regt as an excuse for some of the Wiggy-bashing in this thread is not only weak - the decision was taken elsewhere - but simply adds to the Daily Mail blame game (given that PPrune and ARRSE are frequent DM sources of information).
FWIW, I suspect the troops to task equation was probably, as alluded to by Foghorn Leghorn (post #4), beyond the scale available from the RAF Regt and the task was also seen as airport security, not airfield defence. From a C2 perspective there would have been much less risk in sending a formed battle group than in having to integrate other fighting units, which is perhaps why 16 Air Assault Bde was the preferred option. I would have added an FP HQ to provide specialist airfield ops advice - perhaps someone in the know can tell us if any extra Regt personnel were deployed beyond those already embedded in 16 AAB?