A bigger question is how much conversion would do for a Jew. Quite a lot pre 1890s, but probably much less 1910-1970.
Barry Goldwater comes close to counting as an example of this — he wasn’t a convert, as he was raised Episcopalian, but he was ethnically half-Jewish through his father. Obviously, to say the least, he had a lot of issues which negatively affected his electability, but I don’t think that this was one of them; after all, the handful of states he did win were probably the most anti-Semitic in the country. I suppose one can question how widely this fact about his ancestry was actually known, but his surname is fairly clearly an anglicised German Jewish one.
Not to bog this thread down too much, but I would dispute the characterization of the South as inordinately anti-semitic, even in the 1960s.
There was literally a Jew in the CSA’s cabinet. That would not have been the case if it was really that anti-Semitic.
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