I can prove the following weaker statement, which was not true in the other version:
Every hypergraph stemming from the cliques of a graph has a minimal cover.

The proof follows from Zorn's lemma.
Define a poset $P$ on the covers such that $C<D$ if for every clique $c\in C$ there is a clique $d\in D$ such that $c\subset d$.
$P$ satisfies the conditions of Zorn's lemma, as in any chain $C_1<C_2<\dots$ we can consider all clique-chains $c_1\subset c_2\subset\ldots$ where $c_i\in C_i$, and let $C=\{c\mid c=\cup c_i$ for some such clique-chain$\}$.
A maximal element of $P$, guaranteed to exist by Zorn's lemma, is necessarily a minimal cover.

I can prove the following weaker statement, which was not true in the other version:
Every hypergraph stemming from the cliques of a graph has a minimal cover, where I define a cover as minimal if the union of no two of its hyperedges is a hyperedge.

The proof follows from Zorn's lemma.
Define a poset $P$ on the covers such that $C<D$ if for every clique $c\in C$ there is a clique $d\in D$ such that $c\subset d$.
$P$ satisfies the conditions of Zorn's lemma, as in any chain $C_1<C_2<\dots$ we can consider all clique-chains $c_1\subset c_2\subset\ldots$ where $c_i\in C_i$, and let $C=\{c\mid c=\cup c_i$ for some such clique-chain$\}$.
A maximal element of $P$, guaranteed to exist by Zorn's lemma, is necessarily a minimal cover.