I think you mis-read the sentence. It says “We need more money to keep the Unions, and my special constituents who pay very little or no taxes, happy,...”
Notice the “, and my special constituents who pay very little or no taxes,” is a clause that does not modify the first part of the sentence. He is saying “Keep unions happy” and “Keep the blood-sucking leeches who don’t pay any taxes (i.e., 50% of people who have a paycheck)” happy. The author is referring to the two groups individually, not saying that union members don’t pay taxes.
You’re correct.