It's a really, really good question.
Unfortunately, there isn't a whole lot of information out there. A recent systematic review, one of the highest forms of evidence, was published this year. It said just that: that there isn't a lot of research in this field.
Several studies do report increases in the overall rates of self-initiated death, and in some cases increases in non-assisted suicide in countries that have brought in MAID-type procedures. In particular, women in some of the Benelux countries are increasingly accessing MAID for mental illness. That's a pattern. Typically, women who engage in self-harm choose non-lethal methods to do so. However, there's been a rising number of women with psychiatric illnesses dying from MAID provision for mental illness, which is in contrast to what we see in men.
Certainly some of the trends from Europe are that MAID for mental illness seems to disproportionately affect women, and in some countries it appears to also have a contagion effect in increasing the non-assisted suicide rate. That's something we see in suicide research anyway: There's also a contagion effect when there are suicides locally, and that's why we have very strict media guidelines and reporting about suicide and how it is portrayed in the media.