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The answer is no.

This follows from Kameryn's answer at your other question, and they already basically explained it there. Namely, KM implies the existence of a transitive model of NBG, and transitive models will always satisfy the $\in$-induction scheme. But further, by taking the $L$ of that model, you will get $V=L$ as well, and this implies that there is a definable global well order, allowing you to interpret $\prec$ by the $L$-order. So the theory cannot be equiconsistent with KM, since KM implies the consistency of it.

Incidentally, as Kameryn alluded, many people take the global choice as part of the standard axiomatization of NGB, which is why they had mentioned it.

The answer is no.

This follows from a modification of Kameryn's answer at your other question. Namely, KM implies the existence of a transitive model of NBG, and transitive models will always satisfy the $\in$-induction scheme. But further, by taking the $L$ of that model, you will get $V=L$ as well, and this implies that there is a definable global well order of all the sets, giving us global choice. This by itself does not imply what you call class well order (well-ordering the classes), but since we have a transitive model of $\text{ZFC}+\text{V=L}$, we can also apply the $L$-order to the classes themselves, thereby providing an interpretation of $\prec$ that fulfills your theory.

Putting it all together, it follows that your theory cannot be provably equiconsistent with KM, since KM implies the consistency of it.

The argument is related to how one can realize a certain ambiguity in what V=L should mean in the class context. It is possible in a model of KM that every set is in L, but this is weaker than asserting that every class is constructible in the suitable sense, since one can force over the model to add generic classes, without adding any sets, but these generic classes will not be realized as constructible. However, in any KM model one can go to the inner model $L$ of all constructible sets, and also take as classes only those $X\subseteq L$ that are themselves witnessed as constructible in the (class encoded) constructible hierarchy that proceeds beyond Ord, using classes $\Gamma$ that encode well orderings beyond Ord. In this way, one produces a model of the class choice principle CC, showing that KM is equiconsistent with KMCC. Meanwhile, CC is known not to be provable in KM.

The answer is no.

This follows from Kameryn's answer at your other question, and they already basically explained it there. Namely, KM implies the existence of a transitive model of NBG, and transitive models will always satify the $\in$-induction scheme. But further, by taking the $L$ of that model, you will get $V=L$ as well, and this implies that there is a definable global well order, allowing you to interpret $\prec$ by the $L$-order. So the theory cannot be equiconsistent with KM, since KM implies the consistency of it.

Incidently, as Kameryn alluded, many people take the global choice as part of the standard axiomatization of NGB, which is why they had mentioned it.

The answer is no.

This follows from Kameryn's answer at your other question, and they already basically explained it there. Namely, KM implies the existence of a transitive model of NBG, and transitive models will always satisfy the $\in$-induction scheme. But further, by taking the $L$ of that model, you will get $V=L$ as well, and this implies that there is a definable global well order, allowing you to interpret $\prec$ by the $L$-order. So the theory cannot be equiconsistent with KM, since KM implies the consistency of it.

Incidentally, as Kameryn alluded, many people take the global choice as part of the standard axiomatization of NGB, which is why they had mentioned it.

The answer is no.

This follows from Kameryn's answer at your other question, and he already basically explained it there. Namely, KM implies the existence of a transitive model of NBG, and transitive models will always satify the $\in$-induction scheme. But further, by taking the $L$ of that model, you will get $V=L$ as well, and this implies that there is a definable global well order, allowing you to interpret $\prec$ by the $L$-order. So the theory cannot be equiconsistent with KM, since KM implies the consistency of it.

Incidently, as Kameryn alluded, many people take the global choice as part of the standard axiomatization of NGB, which is why he had mentioned it.

The answer is no.

This follows from Kameryn's answer at your other question, and they already basically explained it there. Namely, KM implies the existence of a transitive model of NBG, and transitive models will always satify the $\in$-induction scheme. But further, by taking the $L$ of that model, you will get $V=L$ as well, and this implies that there is a definable global well order, allowing you to interpret $\prec$ by the $L$-order. So the theory cannot be equiconsistent with KM, since KM implies the consistency of it.

Incidently, as Kameryn alluded, many people take the global choice as part of the standard axiomatization of NGB, which is why they had mentioned it.