The Wikipedia article on Tetragrammaton gives a long list of examples from Greek and Latin in early manuscripts and patristic writing. The overwhelming majority use "Lord", but a few use proper transliterations, such as Ἰαῶ in Greek and "Jaho" in Latin.
Virgil uses magicas artes in Aeneid 4.493:
Testor, cara, deos et te, germana, tuumque
dulce caput, magicas invitam accingier artes.
The adjective magicus seems to be right in the semantic field you're looking for, and it has barbarian connotations baked into the etymology.
Macrobius gives the form of a spell (an evocatio) designed to call forth the gods of an enemy city before attacking it, inviting them to Rome:
si deus, si dea est, cui populus civitasque Carthaginiensis
est in tutela, teque maxime, ille qui urbis
huius populique tutelam recepisti, precor venerorque
veniamquea vobis peto ut vos populum civitatemque
This answer only concerns Latin; I will leave Greek to others.
Vocative is not the way to go here.
It is used for addressing the god, not for such exclamations.
(At least I have never seen it in such use.)
I would swear using pro (or proh) and a nominative.
See, for example, these search results for the second phrase.
In De Civitate Dei, Augustine quotes Varro, who was the author of Antiquitates rerum divinarum. The latter was making distinctions between the various conceptions of the Roman gods. Concerning the third genus of these conceptions, he says:
Tertium genus est [...] quod in urbibus cives, maxime sacerdotes,
nosse atque administrare debent. In quo est, ...
Like English, Latin has words for particular ceremonies, but unlike English, it does actually have a general word, caerimonia, which is used for sacred ceremonies (but not for profane usages). Another word, ritus, is more general and may be used for both sacred and profane.
Recent research has shed some new light on this question. A dissertation that came out of Rutgers in 2007 called the whole idea of ius sacrum into question.
Johnson takes a look at the evidence behind the standard interpretation. For reference, for years everyone relied on Adolf Berger's article in the RE and, later, his entry in Encyclopedia Dictionary of ...
The oldest Greek transcription I've found is from Diodorus of Sicily (The Library of History I.94.2):
παρὰ δὲ τοῖς Ἰουδαίοις Μωυσῆν τὸν Ἰαὼ ἐπικαλούμενον θεόν
Among the Jews, Moses [attributed his laws to] the god called "Iaō".
The oldest Latin one I've found is Pseudo-Jerome (Breviary on the Psalms 8: in this manuscript, it's on 12v-13r):
This is how it is read.
But as for the “sacrificial” tradition, it is not a sacrifice but a votive offering in a tradition still practised in many parts of the Latin-speaking world. You promise God or a saint to make an offering to get you out of a sticky spot. The offering is of apposite form and nature.
For health troubles it might be an eye or an ear ...
I think there probably are Roman deities that the Romans thought were worshipped by Etruscans. I'm less sure about the "specifically" Etruscan part. So far, I haven't seen a primary source that says anything clearly along the lines of "the Etruscans worshipped this god before the Romans did".
A lot of this stuff seems to be unclear, so this is just a ...