$\newcommand\Ord{\text{Ord}}\newcommand\HOD{\text{HOD}}$Let me
point out that the existence of a definable bijection of $V$ with
$P(\Ord)$ is equivalent to the assertion that the universe is
Leibnizian over the ordinals, that is, that any two distinct sets
$a\neq b$ satisfy different formulas with ordinal parameters.

This is proved in Ali Enayat's fine paper, On the Leibniz-Mycielski axiom in set theory, where
he discusses a variety of issues surrounding this axiom.

For the forward implication, suppose that we have a definable
bijection of $V$ with $P(\Ord)$, given by a specific definition.
Thus, to every object $a$ we have definably associated a distinct
set $F(a)\subset\Ord$. In particular, if $a\neq b$, then there is
some ordinal $\alpha$ such that $\alpha\in F(a)$ and $\alpha\in
F(b)$ get different truth values, which establishes that the types
of $a$ and $b$ over the ordinals are different. We could have even
allowed that the bijection was definable with ordinal parameters.

Conversely, suppose that the universe is Leibnizian over the
ordinals. This means that whenever $a\neq b$, then there is some
formula $\varphi$ and ordinal parameters $\vec\alpha$ such that
$\varphi(a,\vec\alpha)$ is true and $\varphi(b,\vec\alpha)$ is
false. By reflection, there is some ordinal $\theta$ such that
$a,b,\vec\alpha\in V_\theta$ and $\varphi$ is absolute to
$V_\theta$. Thus, we have $V_\theta\models\varphi(a,\vec\alpha)$
and $V_\theta\models\neg\varphi(b,\vec\alpha)$. Since $V_\theta$
is $\Pi_1$ definable, this shows that $a$ and $b$ have different
$\Sigma_2$ formulas with the parameters $\varphi,\vec\alpha$ and
$\theta$. (In particular, because we have bounded the complexity
of the formula this way, this shows that being Leibnizian over the
ordinals is first-order expressible in ZFC.) For each object $a$,
let $\theta_a$ be the next $\Sigma_3$-correct ordinal beyond the
rank of $a$, and let $T_a$ be the type of $a$ in $V_\theta$ over
ordinal parameters less than $\theta$. Thus, $T_a$ is the set of
tuples $\langle\varphi,\vec\alpha\rangle$ such that
$V_{\theta_a}\models\varphi(a,\vec\alpha)$. Since any $\Sigma_2$
assertion about $a$ will reflect below $\theta_a$, it follows from
the Leibnizian assumption that $a\neq b$ implies $T_a\neq T_b$.
Further, since $T_a$ is a set of tuples of formulas and ordinals,
we may by Gödel coding view $T_a$ as a set of ordinals. Thus,
we have defined an injective map $a\mapsto T_a$ of $V$ into
$P(\Ord)$, as desired.

In the Leibnizian property, we may fold the ordinal parameters
$\vec\alpha$ into the ordinal $\theta$, by making $\vec\alpha$
definable in $V_\theta$, and say it equivalently like this: for
any $a\neq b$, there is some ordinal $\theta$ and formula
$\varphi$ for which
$V_\theta\models\varphi(a)\wedge\neg\varphi(b)$. This is what
Enayat calls the Leibniz-Mycielski axiom LM.

At that the conclusion of that paper, he conjectures that the
implication $V=\HOD$ to LM is not reversible.