The first difficulty is handled by Scott's trick: given a class $K$, let $\alpha$ be the smallest ordinal such that there are elements of $K$ of rank $\alpha$ (that is, in $V_\alpha$); then we can think instead of the set $K\cap V_\alpha$. This gives a canonical way of describing an equivalence (proper) class, without having to pick representatives.
So Scott's trick lets us replace each equivalence (proper) class with a set. Now we just need to count those sets. Since the powerset of $V_\alpha$ is again a set, it's enough to show that every line bundle has an isomorphic bundle of low rank; specifically, that there is some fixed $\alpha$ such that every line bundle over $X$ is isomorphic to one in $V_\alpha$. But this isn't hard to do; the space $X$ and the field $k$ lie in some $V_\beta$, the underlying set of the total space of a line bundle can be thought of as $k\times X$ (namely: for any line bundle, there's an isomorphic one with total space $k\times X$), and the additional data consists of some maps and some sets, all of which will have rank at most $\beta+3$ (my counting might be off by $1$ but oh well).
In particular, the Picard group can be taken to have underlying set consisting of a bunch of subsets of $V_\alpha$, which are the intersections of equivalence (proper) classes with $V_\alpha$, for $\alpha$ of appropriately high rank.