Let $\mathbb{P}$ be a proper notion of forcing, having the Sacks property. Suppose that $\dot{D}$ is a $\mathbb{P}$-name for an infinite subset of $\omega$. I'm looking for a set which approximates $\dot{D}$ both from above and below, that is:

Is there a set $A\subseteq\omega$ (in the ground model) and a $p\in\mathbb{P}$ such that (1) $p\Vdash\dot{D}\subseteq\check{A}$, and (2) for any finitely many elements $a_1,\ldots,a_n\in A$, there is a $q\leq p$ such that $q\Vdash a_1,\ldots,a_n\in\dot{D}$?

It might be useful to note that if $q_n$ is a decreasing sequence in $\mathbb{P}$ such that each $q_n$ decides whether $n\in\dot{D}$, then $A=\{n:q_n\Vdash n\in\dot{D}\}$ satisfies (2), though I don't think it satisfies (1) if $\Vdash\dot{D}\notin\mathbf{V}$. Meanwhile, the Sacks property ensures that there is a set in the ground model satisfying (1), but at least for the set you get by naively applying the Sacks property, it need not satisfy (2).