If the RN can manage to mount guard without having a full time sqn for synchronised mincing I’m sure the RAF could.There´s also the shiny issue of the Queen´s Colour Squadron, which isn´t a capability that the RAF is going to want to lose. Who is going to man that unit if the RAF Regiment is disbanded? Let´s face it, the likelihood of the Air Force relinquishing their only unit that can mount guard at Buckingham Palace is as high as them disbanding the Sparrows.
They could dick technicians and personnel from other trade groups to do a public duties tour with the QCS, the same way they dick them to stag on at the gates in lieu of RAF Police, only a public duties tour will be for three years, and not just a for weekend. I can´t see that being too popular with the average crab.