I think that will work. One possible construction is to pick $\mathfrak{c}$ many countable proper subsets of $\mathscr{P}(\mathbb{N})$ indiced by real intervals of the form [x,y], where x,y \in $(i, j]$ with i,j consecutive integers starting from 1. That is
$\{\forall x,y \in (\infty,1]|S_{[x,y]}\cup \{0\}\}$. Then for any two countable subsets indiced by consecutive intervals [i,i+1] and [i+1,i+2], the intersection is guarenteed to be the singleton i+1. If the pair are not consecutive intervals, then the intersection is guarenteed to be the singleton {0}