Let's take $\sf MK$ set theory. 

Adopt the notation of upper case ranging over all objects, lower case only range over sets (i.e.; elements of classes), and $\frak A,B,C,..$ to range only over proper classes (non-sets).


 *Define:* $X \approx Y \iff \forall m \, (m \in X \leftrightarrow m \in Y)$

Weaken Extensionality to:

**Class Extensionality**: $\forall X \exists! {\frak A} : X \approx \frak A $ 

**Set Extensionality:** $\forall x \exists! y : x \approx y$

Modify limitation of size to:

**Limitation of size:** $|X| \neq |V| \to \exists x: x \approx X$

Where $V$ is the class of all sets, and $||$ stands for cardinality function.

Add all other axioms of $\sf MK$.

Call this theory $\sf MK'$. 

So, $\sf MK'$ have an Ur-proper class for every set.

> Is $\sf MK$ bi-interpretable with $\sf MK'$?

I think they are equi-interpretable! The proof is: 

Define a new membership relation $\in^*$ by: $$ Y \in^* X \iff \exists z: z \in X \land Y \approx z$$

I think that $\sf MK'$ would prove all axioms of $\sf MK$ with $\in$ replaced by $\in^*$ and $=$ replaced by $ \approx $. 


For the other direction, we need to define a new membership relation $\in'$ along the following lines:

*Define:* $\operatorname {binarytuple}(X) \iff \\  X: V \to \{0,1\} \land \\ \exists x \forall y (y \in x \leftrightarrow \langle y, 1 \rangle \in X)$

*Define:* $Y \in' X \iff \\ \operatorname {binarytuple}(X) \land \langle Y,1 \rangle \in X \lor \\  \neg \operatorname {binarytuple}(X) \land Y \in X$

Accordingly, $\sf MK$ would prove that for every set there is a unique proper class that is $\in'$-coextensional with it. And so, $\sf MK$ would prove all axioms of $\sf MK'$ with $\in$ replaced by $\in'$.

Of course, this doesn't prove them [bi-interpretable][1]. Hence, the question raised above.

 


  [1]: https://en.wikipedia.org/wiki/Interpretation_(model_theory)#Bi-interpretability