Usually the notation WO refers to the well-order principle for sets only, and this is usually taken as part of KM. So let me refer to your global-well-order principle as the class-well-order principle CWO, since your relation $\prec$ is placing a well-order on classes.
First, I claim that that CWO is not provable in KM, in the sense that we can have models of KM that have no second-order definable relation $\prec$ fulfilling CWO. One way to produce such a model is to start with an inaccessible cardinal $\kappa$ in ZFC and then force to add Cohen subsets to $\kappa$ and build a symmetric extension $V\subseteq W$ such that the axiom of choice fails in $W$ for subsets of $\kappa$. But since no sets were added below $\kappa$, we still have KM in the structure $\langle V_\kappa,\in,P(V_\kappa)^W\rangle$. In this model, we have KM, including global choice, but no definable well-ordering of the classes, since there isn't even such a well-order in $W$, and so definable CWO fails. Furthermore, inside $W$ there is no way to place a relation $\prec$ on this KM model so as to satisfy CWO. I believe that one can omit the need for the inaccessible with a more careful argument over KM itself.
Second, I claim that your proposed axiom involving $\varepsilon\phi$ does not imply CWO, and indeed, your proposed axiom is already provable in KM. Your proposal is that $\varepsilon\phi$ chooses an element of the definable class $\{x\mid \phi(x)\}$. Thus, you have a global choice function that operates on classes, not just on sets. But this is equivalent to global choice, since if $F$ is a global choice function on sets, so that $F(x)\in x$ for every set $x$, then we can also choose from classes, since we just apply $F$ to the minimal-rank elements of any class. That is, we can define $\varepsilon\phi$ to be $F$ applied to the minimal-rank elements of $\{x\mid\phi(x)\}$. Thus, your axiom is true in every model of KM. (See my blog post [The global choice principle in Gödel-Bernays set theory](https://jdh.hamkins.org/the-global-choice-principle-in-godel-bernays-set-theory/) for further various equivalent formulations of global choice.)
[Update] You have clarified in the comments that you intended the proposed class-choice principle is picking classes from definable collections of classes. Thus, for any formula $\phi(X)$ with a class variable, if there is any class realizing it, then $\varepsilon\phi$ is a class realizing it.
This principle is of course a consequence of CWO, since one can take the $\prec$-least instance. And conversely, one might hope to prove from that axiom that CWO holds, by iteratively choosing classes along a meta-class order, in the same manner that Zermelo proves the well-order principle from the axiom of choice.
That method, however, will not work in all models of KM. For example, if $\kappa$ is inaccessible in ZFC and $2^\kappa=\kappa^{++}$, a situation one can arrange by forcing, we may consider the corresponding KM model $\langle V_\kappa,\in,P(V_\kappa)\rangle$. In this model, there are only $\kappa^+$ many meta-ordinals, but $\kappa^{++}$ many classes, and so the iteration will not succeed in placing all the classes in order.
We can certainly augment this model with an interpretation of $\varepsilon\phi$, choosing a class from every definable meta-class, so as to realize KM plus your proposed axiom, but it is not yet clear to me whether we can do so without having any class well-order $\prec$ definable from it. I believe this is possible.
Meanwhile, let me also point out that your proposed axiom implies the class-choice axiom CC [see article "Kelley-Morse set theory does not prove the class Fodor principle" end of page 4 and beginning of page 5], which asserts that if every set $a$ has a class $A$ with $\varphi(a,A,Z)$, then there is a class $X\subseteq V\times V$ for which $\forall a\ \varphi(a,X_a,Z)$, where $X_a$ is the $a$th section of $X$. This axiom is known not to be provable in KM, although KM+CC is interpretable in KM as I explain below. The point is that with your principle, we can form $X$ by using $\varepsilon\varphi(a,\cdot,Z)$ to place the chosen witness $A$ for which $\varphi(a,A,Z)$ on slice $a$.
Lastly, you ask about interpretation, and indeed KM can interpret a model of KM+CWO, in which $\prec$ is second-order definable. That is, we don't need any choice operation $\varepsilon\phi$ at all to interpret KM+CWO. The reason is that if KM holds in $V$, then we can interpret the constructible universe $L$, and furthermore, we can take as classes only those that are witnessed as constructible at a meta-ordinal stage represented by a class in $V$. That is, using the classes of $V$, we can represent "ordinals" beyond $\newcommand\Ord{\text{Ord}}\Ord$ by well-founded class relations on $\Ord$. And by the standard coding techniques using well-founded extensional binary relations on $\Ord$ to code "sets" of rank above $\Ord$, we can refer to the stages of the constructible universe above $\Ord$. Let us take the model $L$ as constructed in $V$, using classes those that are witnessed as constructible at some such meta-ordinal stage. One can show that this satisfies all the KM axioms, and furthermore, there will be a definable well-order of the classes in this model arising from the $L$ order on the meta-ordinal stages.
This method is the main method used to show that KM is equiconsistent with KM+CC. It has appeared in several papers, and there is a very nice account of this class-coding method in the dissertation of my student Kameryn Williams.
