I agree that the phrase is used adverbially in your sentence, not adjectivally. In any case, it is a participial phrase.
In the earlier days of transformational grammar, it would certainly have been taken as a reduced clause, deriving from the underlying sentence "John kissed Atsuko." This clause would first have undergone certain so-called complementizer transformations, giving, in this case, an intermediate structure something like "because of John having kissed Atsuko." This would then be passivized and pronominalized, yielding "because of her having been kissed by John." Note that the auxiliary verb "have" would be required here to indicate that the kissing is prior to the fainting. Equi-NP deletion would have gotten rid of the pronoun "her" under identity with the main clause subject Atsuko, and at that point we would have a possible surface structure like
Because of having been kissed by John, Atsuko fainted
Further optional reduction and movement would give your sentence.
I don't see why you have a problem with the placement of the adverbial, either as a full clause or a reduced one. There are three places it can go when attached to the main clause, "Atsuko fainted," and it could, in fact, readily appear in all three.
References to the kind of derivation I've outlined above are scattered throughout the literature of early transformational grammar, including books and papers by Chomsky, Emmons, Bach, Rosenbaum, and many others.
More recently, Chomsky has proposed refinements of his theory of syntax which cast considerable doubt on whether it is desirable at all to derive such phrases from full clauses in this way. Instead, he suggests that the verb "kissed" will at some level be assigned the feature "passive," that its subject will be construed as John and its object as Atsuko, without any need to posit the perhaps illusory existence of a deep structure clause underlying the participial phrase of which it is the head. The English verb "kiss" simply must be assigned an agent and a patient, and in the structure we have, these must be as I've stated. So my moral here is that it is probably best not to worry overmuch whether "kissed by John" is really a reduced clause or not. It is, all sides agree, even traditional grammarians, a participial phrase used adverbially, and the verb at its head requires that it be interpretable, in context, with the subject and objects it in fact does have.
