If Saddam wanted to go off about international relations he'd embarrass France and Russia a lot more than he'd embarrass us.
And please, let's not dignify our relations with Saddam in the 80s as some kind of alliance. if we were allied with Saddam, we were allied with 100 other countries. Man, that Reagan, hell of a diplomat that guy.

In actuality, Iraq was always a Soviet client state.
Saddam was the enemy of our enemy, nothing more. We weren't the only ones who saw it that way. Pretty much the entire world sided with Saddam against Iran, which was seen as the bigger threat at the time and in hindsight probably was.
As for the gassing of the Kurds, that atrocity resulted in international commendation and pretty much ended Saddam's good relations with most of the Western world.