Injection of the proper class of ordinals in every proper class

Is it possible to prove in the set theory NBG (with local choice but without global choice) that the proper class of ordinals injects in every proper class ?

To see this, consider the class $W$ consisting of all well-orderings of any rank-initial segment $V_\alpha$, for any $\alpha$. If we had an injection of Ord into $W$, then there must be unboundedly many $\alpha$'s that are used, since each $V_\alpha$ has only a set-sized family of well-orderings. Thus, we have a global selection of well-orderings of unboundedly many $V_\alpha$, and from this we can define a well-ordering of the entire universe. Namely, $x<y$ if the rank of $x$ is lower than that of $y$, or if they have the same rank and $x<y$ in the first well-ordering of some sufficiently large $V_\alpha$ to appear in the range of the injection of Ord into $W$.