By RAW a hermetic virtue cannot be gained by a pilgrimage, but I am wondering how many would consider an exception for gentle Gift. It would certainly make sense if the social penalty for the Gift is seen as a divine penalty or limitation.
Alternately how would you go about creating an initiation ritual for the gentle Gift that might appear to be a pilgrimage.
I would probably have a large amount of travel and interaction with mundanes. Probably do an initiation ritual similar to the Jerbiton Itinerarium.
If you're trying to make it more like a holy pilgrimage, travel to a number of populous holy sites as a humble traveller, symbolizing casting aside the hubris of magic and the social penalty of the gift. I'd also take humble or vow as an ordeal, and throw in a lot of monetary donations.
A good vow would be a vow to use ones magic to improve the lives of the poor and the humble folk.
Gentle Gift is one of those virtues that should be named something else rather than hermetic, because anyone with the Gift can have the Gentle Gift, not only hermetic magi. It makes sense to me that you can't get Puissant Ignem from a pilgrimage, but I see no inherent problem with gaining Gentle Gift.
Still, I'm not sure I'd allow it as a default option. And personally I don't like the possibility of gaining it through an initiation either (the cults that can initiate GG in the books either are not canon in my games or don't have this particular script).
If you must have an initiation akin to a pilgimage, I think sacrifice of personal time (the travel) to a holy site of significance, in an auspicious time of the year, fits the bill very well. Maybe with the effects tied to a particular relic. Eg., after traversing the way of Santiago your Gift is gentle, but only while you wear the St. James shell around your neck.
This angers the ash guild...
It probably does, but if you care enough about what the ash guild is angry about then going on a pilgrimage to get the Gentle Gift is probably not for you anyway.
And likewise the divine is unlikely to take the preferences of the ash guild in consideration for who it grants the Gentle Gift to and what it demands in return. 
Which also brings up an interesting thought- could an unGifted mystagogue help Gifted supplicants initiate hermetic virtues? For example a priest with a high presence and mystery cult lore guiding them through initiation(hopefully also unaffected by the Gift)?
in the way you phrase the question I would say: probably not.
Specifically I dont think that the divine would grant virtues using the methods that hermetic magi use to do it on their own.
More likely IMO the divine would grant virtues (hermetic or not) to magi according to the same method it grants virtues to non-magi.
Though I think it very likely that the divine could and in rare cases would grant virtues to magi deemed to be needing those virtues, and that it is likely that a priest may be involved in the process in some way.
So in a sense I would say that yes what you describe can probably happen but probably not in the way you describe it.
I think you have misread something, because at no point have I asked about the divine granting virtues through initiation. I have asked about an initiation which mimics the divine, and I have asked about unGifted people who might have an affiliation with the divine performing initiations, but neither of these involves the divine granting virtues. The question of the divine granting Gentle Gift through pilgrimage is a separate but related question on whether gentle Gift is an appropriate virtue to e granted with divine methods.
In that case I believe that the answer is simply, no, they cannot.
From my reading of The Mysteries and Hedge Magic (in many senses there is little difference between a mystery cult and a tradition) it seems that yes, generally speaking a non-Gifted member can initiate a Gifted member. There is also no problem in initiating a virtue not know by the mystagogue (except a few cases, eg., folk witches must know the virtue they wish to initiate).
I don't think there is a clear answer, RAW, but it seems to be possible. But from a narrative point of view, I think it would depend on the virtue.
Personally, if it's something "too hermetic", eg. Deft Form, I would probably say no. If it's something that arguably a non-hermetic mystagogue could have some degree of understanding even if he doesn't know the virtue, eg. Cyclic Magic (Positive), I'd say it's reasonable to initiate the virtue.
You could argue that the script is just a recipe, and as long as the mystagogue follows the recipe it really doesn't matter if he understands what he is doing (and so a non-hermetic mystagogue could initiate any kind of reasonable hermetic virtue). I think this would also be fine.
Here's what I missed in my earlier conjecture- An unGifted character cannot act as the
Mystatogue unless he knows the power that the Initiation develops — this
means he cannot Initiate himself (from hedge magic)
Which means an unGifted Mystagogue cannot initiate anyone with the Gentle Gift, since they cannot have the virtue.
Though if it is a covered trait of a tradition I suppose you could learn it from a book. Which is just weird as a general concept.
Learning from books (or a regular teacher) only works for [Supernatural] Abilities, not virtues without a corresponding ability.
In Apprentices they have rules for teaching advantages, but the teaching is penalized for what abilities you already have. Tradition abilities (and presumably virtues) do not suffer those penalties.