It is conceptually simpler if you work in a class theory such as NBG or MK, in which classes are actually objects. However, if you want to see how the issue looks like in ZFC, you have to always remember that a class is nothing more than a $1$-parameter sentence over ZFC. For example, the Russell class $R$ would be defined by $R(x) \equiv x \notin x$. And an arbitrary collection is nothing more than a class. Similarly, a bijection between classes $C$ and $D$ is nothing more than a $2$-parameter sentence $F$ over ZFC such that ZFC proves "$\forall x \in C\ \exists! y \in D\ ( F(x,y) ) \land \forall y \in D\ \exists! x \in C\ ( F(x,y) )$". Note that what I just wrote can be expanded to a sentence over ZFC. In other words, classes are definable properties over ZFC, and bijections between classes are definable functions that are provably bijective. Finally, a proper class is a class $C$ such that ZFC proves "$\neg \exists S\ \forall x\ ( C(x) ⇔ x \in S )$". For instance, $R$ is a proper class since ZFC proves "$\neg \exists S\ \forall x\ ( x \notin x ⇔ x \in S )$".
With this in mind it is easy to see that the claim for ZFC is essentially a claim in the meta-system:
For any classes $C,D$ and bijection $F$ from $C$ to $D$, if $C$ is a proper class then $D$ also is.
Proof sketch: Work inside ZFC under the assumption that $\exists S\ \forall x\ ( D(x) ⇔ x \in S )$:
Let $S$ be such that $\forall x\ ( D(x) ⇔ x \in S )$.
By replacement and bijectivity of $F$, let $U$ be such that $\forall y \in S\ \forall x \in C\ ( F(x,y) ⇔ x \in U )$.
Then by bijectivity of $F$ again, we have $\forall x \in C\ ( x \in U )$.
Thus by specification $\exists T\ \forall x\ ( C(x) ⇔ x \in T )$.
Therefore ZFC proves that $\neg \exists T\ \forall x\ ( C(x) ⇔ x \in T ) ⇒ \neg \exists S\ \forall x\ ( D(x) ⇔ x \in S )$, and the claim follows.
This should also address the issue of how we "automatically know the claim" despite the contradiction being 'specific' to the proper class. Namely, the claim is a meta-theorem that essentially gives us a constructive way to convert any proof that $C$ is a proper class into a proof that $D$ is a proper class, given any proof of a bijection between $C$ and $D$. Yes, the proof generated will vary according to the proofs provided, but as shown above this conversion works uniformly.