Regarding your question of the relation between CT and omniscience principles, I think this needs to be a separate question (and you will need to make clear exactly what you mean by CT because there are several variants). Let me note that, if I am not mistaken, Lifschitz realizability validates some form of CT (namely: “every function from naturals to naturals is Turing-computable”) and also LLPO (analytic and sequential; note that Cauchy and Dedekind reals coincide in Lif), so at least (some form of!) CT is compatible with LLPO.
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