Every nonempty finite set $x$ has an $\in$-maximal element, a set $z\in
x$ with $z\notin u$ for any $u\in x$, since one can simply climb up via $\in$ inside $x$ until one
reaches a maximal element.

But the statement is not true of all infinite sets, since the set
HF of hereditarily finite sets itself, for example, has no
$\in$-maximal elements.

Using this idea, we can produce a counterexample to your transfer principles. Let $\phi(y,x)$ assert that $x$ is not empty and
if $x$ has an $\in$-maximal element, then $y\in x$. This can be
expressed by a comparatively short formula. Namely, let $\phi(y,x)$ be the assertion $$(\exists z\
z\in x)\wedge[(\exists z\in x\ \forall u\in x\ z\notin u)\to y\in x].$$

If $x\in\text{HF}$ and $\phi(y,x)$, then $x$ is not empty and $y\in
x$, since every nonempty element of $\text{HF}$ has $\in$-maximal
elements. But there are sets $x$ in $V$ which are not empty, but
have no $\in$-maximal elements, such as $x=\omega+\omega$, and in
this case $\phi(y,x)$ holds for all $y$, including the case where $y$ is a proper class.

So this violates your transfer principles.