As discussed in Noah Schweber's answer to What is the proof-theoretic ordinal of true arithmetic?, it is somewhat ambiguous what “the proof-theoretic ordinal of True Arithmetic” might mean. In one sense, it is $\omega_1^\text{CK}$, because e.g. True Arithmetic satisfies $\Sigma^0_1$ induction for every notation for a recursive ordinal. But of course $\Sigma^0_1$ transfinite induction is not the same as full transfinite induction, which can’t even be expressed in the language of first-order arithmetic, so I’d like to ask about a second-order version of this notion.

My question is, which ordinals $\alpha$ can $\text{ACA}_0$ + True Arithmetic prove are well-founded, in the sense that there exists some arithmetically definable well-ordering $R$ with order type $\alpha$ such that $\text{ACA}_0$ + True Arithmetic proves $\operatorname{WO}(R)$? Where $\operatorname{WO}(R)$ is the $\Pi^1_1$ sentence asserting $R$ is well-ordered, i.e. $R$ satisfies transfinite induction for all sets, not merely for e.g. $\Sigma^0_1$ sets.

Is the answer all recursive ordinals? If so, how does that square with the work of Feferman and Schutte, who made infinitary deductive systems for ramified second-order arithmetic containing $\text{ACA}_0$, the omega rule, and various comprehension schemes, and found that the proof-theoretic ordinal of the resultant systems is $\Gamma_0$, the Feferman–Schutte ordinal which is recursive?