I agree that the relationship is separate question. I did find some links though:

Here is a Swedish conscripts study, finding that pre-morbid IQ was negatively associated with later adult depression, anxiety, and schizophrenia, but positively related to mania, measured by hospital admittance. This New Zealand study replicates this: Low childhood IQ predicts depression, anxiety, while higher IQ predicts bipolar.

These are about the best "large homogeneous" population studies I could find, in two more-or-less standard Western cultures. There is one study that tracked some particularly high performing children through adulthood, but the results weren't much different regarding mental illness than a normal high intelligence sample would be. Needless to say, it gets complicated when you look at populations that are preselected (college students, etc) or more diverse. Most popular articles that claim a uniform association are looking at some narrow populations (e.g. famous artists), or reporting how intelligence relates to different presentations of a given mental illness (e.g. intelligence seems to the presentation of anxiety).

Even assuming genetic risk for a mental illness was unrelated to education or intelligence, you'd expect something like this given the environmental correlates: Better family conditions early on, better social status later. While there are some environmental stressors that are probably associated with higher intelligence (graduate/medical/law school, perhaps more status anxiety?), these are probably not severe enough to outweigh the stressors in the opposite direction.