Let me explain that argument that I had in mind. 





The claim is that if a potentialist system has arbitrarily large
finite families of independent unpushed buttons at every world,
then it's modal validities (in the language with the buttons) is
contained within S4.2.

Your example system does not satisfy the hypothesis, since at the
top world all buttons are pushed. And indeed, your statement Top is
not even provable in S5, let alone S4.2.

To prove the claim, the main idea is that we don't really need full
switches in the main button+switch result that Benedikt Löwe
and I proved in our modal logic of forcing paper; rather, it suffices for the switches to be usable a sufficient finite number of times.
Consider therefore a weakening of the switch concept, which is a
switch that can be flipped a certain number of times.

Specifically, we define recursively a notion of $k$-independence
for a family of buttons $b_j$ and statements $s_i$ to be used as
weak switches. When $k=0$, we say that every such family is
$0$-independent at every world. The family is $(k+1)$-independent
at a world, if you can operate the buttons and switches to achieve
a new configuration (pushing any desired additional buttons and
setting the switches as desired) by moving to a world where the
family remains at least $k$-independent. Thus, a family is
$k$-independent at a world, when you can operate the controls as
you like $k$ times.

Now, following the usual button+switch proof, if a statement
$\varphi$ is not in S4.2, then it fails in a Kripke model $M$ whose
frame is a finite pre-Boolean algebra. We can use the buttons and
weak switches to label the nodes in this Kripke model, just as
before. Every world $w$ in $M$ get assigned a Boolean combination
$\Phi_w$ of the buttons and switches, in such a way that pushing a
button means moving to a higher cluster, and changing switches only
means moving within the cluster. Every propositional variable $p$
is translated to $\psi_p$, the disjunction of $\Phi_w$ for which
$p$ is true at $w$ in $M$.

Now, we prove by induction on modal propositional assertions
$\varphi$ of depth at most $k$, that if the buttons and switches
are $k$-independent at $W$ in the potentialist system $P$, then
 $$W\models_P\varphi(\psi_{p_0},\ldots,\psi_{p_n})\iff
 w\models_M\varphi(p_0,\ldots,p_n),$$
where $w$ is the world in $M$ for which $W\models\Phi_w$. The point is that the induction step requires you to use up only
one life. This shows that we don't need full switches, but only
$k$-independent switches.

The proof is completed by observing that if you have sufficiently
many unpushed buttons, you can form these $k$-independent families.
Fix any $k$, and suppose that we have a very large family of
independent buttons available. Form them into disjoint groups of
size $k$, and for each group, let the corresponding weak switch $s$
be the parity count of the number of them that are pushed in that
group. And then take as many additional of the buttons as you want.

The point is that you can change the parity count of the button
groups simply by pushing one more button in that group. So if
button groups have size $k$, you will achieve $k$-independence.