I've contacted a good friend in India who's an authority on Indian badges and insignia, and when I asked him about the AF and especially their possible button, this was his reply:

'Regarding Aden Rifles, I have not come across a button to this unit with a specific regimental device. The AR were a small unit (just about a coy strength or under), raised in response to the Great War. As you would know during the great War period the AFI units were numbered to their intense dislike, numbers were later dropped in 1920, and I presume the 45th would have become simply the Aden Rifles.

Some AFI units that did not wear specific regimental buttons usually carried a button with a scalloped edge with a Tudor crown in the middle or plain ball or half domed buttons.'

I have since sent him the update of Quicksilver's 65th AR's shoulder titles.
Rgds,
button_guru