What should one do when saying שנים מקרא ואחד תרגום if one comes upon a verse where the Targum (Aramaic translation) has two alternative wordings? I would just say the verse in Targum twice (or four times when there are two textual changes) to reflect both ways; however, I am apprehensive when Hashem is in the verse.
I doubt it would matter whether the verse contains Hashem's name, because when you say an entire verse (in the original Hebrew, or in the Targum), there is no concern about saying His name in vain - no matter how many times you repeat the verse.
In any case, there is an opinion that you can substitute Rashi's commentary for the Targum (Shulchan Aruch, Orach Chaim 285:2). According to that view, I would suppose that there's no reason to repeat the Targum with the variant version. But even according to the opinion that Targum is preferable since it was given at Sinai (Taz ibid. :2), isn't it still part of the Oral Torah? And in that area the exact wording is not as significant (just as when we learn Gemara, we don't need to seek out and repeat every textual variation), and so I'd think that it ought to be fine to just pick one of the alternatives and recite that.
First, it would be well-near impossible to learn Shnayim Mikra in this manner. Here is Shadal's Ohev Ger. Here is R' Avraham Berliner's work on Onkelos. Both list (and Shadal discusses in some detail) the various textual variants which occur. Do you have any idea how long it would take to get every textual variant in Onkelos which occurs?
Alex's suggestion, above, is good. (Though tangentially, look at the sefer Yosef Daas for a discussion of all sorts of variant girsaot of Rashi.) The point is to say it in Aramaic as a means of understanding the Hebrew, and so long as you recite the text before you, I don't think you could be expected to do more.
When you see Hashem's name in the verse, do you pronounce it in Onkelos? I've heard some say that when doing the Shnayim Mikra (=Hebrew) aspect of it, you are not fulfilling it unless you say Hashem's name, 'Adonai', rather than 'Hashem'. (Also, see here.) I am not so convinced that this is so.