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I don't think 50-55% is a stretch for Sicilian-Levantine/Caucasus. It might be pulled towards the Levant but as you mention, there is the Caucasian influence 'weighting' the admixture. Sicilians score as much WHG as AJs - some of both groups score non-trivial amounts - but if you're 50% Sicilian you're not 50% European, Sicilians also have heavy Anatolian/MENA admixture.
A half-Sicilian, quarter-Lebanese quarter-Iraqi Kurd would look pretty much like an AJ, I think. ~50% Sicilian/South Italian and that sort of area + ~25% Judaean + ~25% south Caucus/north Iraq (pre-Arabic Iraqi populations) is for me a satisfactory look at the core ancestral population of Western European Jews. I think it satisfies all genetic and historic criteria. On top of that we have SSA for both, but particularly the SJs, and later, minor, other-European admixture+mongoloid for AJs.
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