What is the ruling on saying "Ya Allah, Ya Allah" to ask for permission, without intending it as a form of remembrance (dhikr), and is it considered forbidden (haram) as has been transmitted from some scholars?

This method of seeking permission is disliked (makrooh) and should not be used, out of reverence for the remembrance of Allah Almighty and the preservation of His Name. What some people do by saying "Subhanallah" (Glory be to Allah) or "La ilaha illa Allah" (There is no god but Allah) when seeking permission is a blameworthy innovation (bid'ah), due to the disrespect it entails towards Allah by using His Name for seeking permission. This is unlike clearing one's throat or knocking on the door. Everything that is legislated as an act of worship to Allah Almighty must only be performed as an act of worship to Him, in a manner of glorification and reverence, not in a manner of triviality.