It is definitely valid translation of epiousios as "necessary for existence", as well as "of transcendant existence". What happens here is that the pedestrian meaning becomes the dictionary meaning, while the outworldly mystical connotation becomes secondary. The exact same thing happens in Russian, "suschnost" being "essence" but "nasuschny" commonly is understood as "vitally necessary".
"What happens here is that the pedestrian meaning becomes the dictionary meaning, while the outworldly mystical connotation becomes secondary."
The mystical connotation doesn't appear in the Liddel and Scott entry at all. With the specific patristic discussions that include "necessary for one's essence" (which is a different meaning from "daily") that amount to a specific discussion of the meaning of the word, it would seem to me to be an oversight on the editors' part not to have given one of those citations -- but there is nothing particularly transcendent about those.
With regard to the "of transcendant existence" meaning to which you refer, I have to say that I saw nothing in my review of the patristic literature available to me (I electronically searched the entire Eerdman's set for references to that line in the Lord's Prayer and reviewed St. Theophylact) that said that this word intrinsically or etymologically means "of transcendent existence" or anything of the like. There are discussion of its deeper spiritual meaning, but this is quite a different thing.
What I found was that the meaning of "daily," "day by day," necessary, etc... referring to minimal physical needs is all over the place as a plain meaning. I also found that there is at least one discussion of the meaning as "necessary for the essence," in St. Theophylact. So these are the only discussions of the plain meanings of the word itself that I found in the patristic literature that I reviewed.
I agree that a transcendent aspect is a spiritual meaning that can be inferred, but I don't see anything in the patristic literature that spells this out as being intrinsic to the word itself.
Your original point to Harley seemed to me to be that a simple look at the word "epiousion" in the Our Father should bring one straight to a Eucharistic or transcendent meaning. My immediate gut level reaction was that the patristic literature wouldn't support your assertion, congenial though it might be, and I've found nothing in my brief review to contradict that gut-level reaction. I'm happy to have this fleshed out, though.
I still do, though, want Harley to tell me how St. Augustine could be so right, by his view, on things like predestination and monergism, and yet so terribly wrong in his belief that included in the Lord's Prayer is an implied command to commune daily!