Note that Emil Jeřábek's answer also provides a negative answer to the second question of whether the consistency of some large cardinal axiom implies the existence of an $\omega$-model of that axiom, since consistency in $\omega$-logic (equivalent to the existence of an $\omega$-model via the Henkin–Orey completeness theorem) strictly implies $\omega$-consistency.

You could also look at §3.7 of Soare's 2016 book Turing Computability; the low basis theorem is theorem 3.7.2. A nice survey is also given by Diamondstone, Dzhafarov and Soare, '$\Pi^0_1$ classes, Peano arithmetic, randomness, and computable domination' (NDJFL 51(1):127–159, 2010).

The set of (indexes of) recursive ordinals, aka Kleene's $\mathcal{O}$, is $\Pi^1_1$-complete, so you would need an oracle for a $\Pi^1_1$-complete set.

@NoahSchweber I don't think $\mathsf{WKL}_0$ is conservative over $\mathsf{RCA}_0$ for $\Sigma^1_1$ sentences. $\Pi^1_1$ sentences yes, but for $\Sigma^1_1$ sentences an instance of WKL for a recursive counterexample like the Kleene tree will be a $\Sigma^1_1$ sentence that $\mathsf{RCA}_0$ cannot prove.

It seems like a proof of this sort would just be pushing the fixed-point lemma into the Gödel numbering. See for example Appendix A of Halbach and Visser (2014).

I would imagine that one could prove this result from an appropriate constructive version of the Hahn–Banach theorem, e.g. the one in §9.3 of Bishop's 1967 book Foundations of Constructive Analysis.