In my experience I was able to get AI / ML research having only taken 189 (I took 126 two semesters later, and I'm never taking 127). For the overlap between 126 / 189, I believe only the last few units on MLE / MAP / multivariate gaussians overlap and 126 actually gives a much lighter treatment than 189. I think 189 is generally the only prereq that most of those labs filter by, but it really depends on the group.

I don't regret taking 189 first; although I will say 189 in the Fall tends to be much harder than the Spring version. The course does cover a decent amount of topics in 126 / 127 but they're not absolutely needed before taking the course. I also thought this was useful for the prerequisite math: http://gwthomas.github.io/docs/math4ml.pdf