minimizing the summed cardinality of set unions
Your problem is NP-complete. We show this by reducing Partition to it.

In Partition you are given a set of numbers $a_1, \ldots, a_k$ and have to decide, if this set can be partitioned into two sets that sum up to the same value.

We map an instance of Partition to your problem by introducing for each $a_i$ the set $(i, a_i, \{1\})$. Set the maximum size to $\frac 12\sum_{i=1}^ka_i$. The original instance can be partitioned if and only if the resulting one can be united to cardinality sum 2.