It seems more likely that the issue is the latter. The Atlantic had a column headlined:
“unions hate private equity, but they love its profits.”
However as the Economist points out (cited in the update), it is not clear that limited partners, like pension funds, actually do better investing in profit equity than investing in stock index funds. There may be an issue with specific officials getting kickbacks from private equity funds, but it is not clear that unions in general would be happy about private equity funds giving them returns that trail major stock indexes.