Solving a variant of the Exact cover problem
There's no pseudopolynomial algorithm for exact cover (unless P=NP), so it is easy to show that this implies there is no pseudopolynomial algorithm for twice-exact-cover, either.

For instance, here's an easy reduction.  Given an instance $(S,X)$ of exact cover (where $S$ is a collection of subsets of $X$), let $X' = X \cup \{a\}$ where $a \notin X$ is a new symbol, and $S' = S \cup \{X', \{a\}\}$.  Now there is a twice-exact-cover for $(S',X')$ iff there is an exact cover for $(S,X)$.  So, if you could solve twice-exact-cover in pseudopolynomial time, you could solve exact cover in pseudopolynomial time, too.