Given classes $H$, $R$, can we find a class $G$ such that the following statement is provable in ZFC?
If $R\subseteq V\times V$ and $H$ is a function from $V$ into $Ord$ such that for all sets $x,y$, $y\ R\ x$ implies that $H(y)<H(x)$, then $G$ is a function from $V$ into $V$ such that for all sets $x$, $G(x)=\{G(y)\ |\ y\ R\ x\}$.
Note that the relation $R$ in the above statement must be well-founded, but need not to be set-like.
The intent of this question is to investigate the relationship between the following three properties of a class relation $R$:
(1) There is a function $H$ from $V$ into $Ord$ such that for all sets $x,y$, $y\ R\ x$ implies that $H(y)<H(x)$, i.e., there is a homomorphism from $[V,R]$ into $[Ord,\in]$.
(2) There is a function $F$ from $V$ into $Ord$ such that for all sets $x$, $F(x)=\mathrm{sup}^{+}\{F(y)\ |\ y\ R\ x\}$, i.e., there is a rank function for $R$.
(3) There is a function $G$ from $V$ into $V$ such that for all sets $x$, $G(x)=\{G(y)\ |\ y\ R\ x\}$, i.e., there is a Mostowski function for $R$.
It is easily seen that (3) $\Rightarrow$ (2) $\Rightarrow$ (1), and Schweber's answer below shows that (1) $\Rightarrow$ (2). The question now is whether we can prove that (1) $\Rightarrow$ (3).