I would not claim to be any kind of expert on mental health, and I think the topic of men and women and mental health deserves a thread in its own right. That said, I would suggest it is the case that preventing deaths due to mental health issues requires interventions at earlier points in the person's life.
There are behaviours that are far more likely to lead to death that women more frequently carry out than men. People with anorexia nervosa are more likely to die from their illness than people with any other sort of mental health disorder. If people exhibit symptoms of anorexia, people around them and mental health services are likely to intervene and think some of that person's choices should be decided with others or sometimes solely by others.
Two issues that are not in themselves mental health issues, 'self harm by ingestion' and 'problematic drug use' are not in themselves cause to section somebody, but they can lead to intervention and being sectioned if there are other accompanying health problems like depression.
Taking harmful substances is usually referred to as 'self harm' if the person is a woman and chooses substances associated with women (over the counter or prescription medications that contain cocaine, sedatives etc, hosuehold cleaning products). It is commonly seen by the public as a 'cry for help' and by professionals as a 'coping mechanism.' This remains the case even when the woman claims it is for adrenalin rush, physiological impact etc. There will usually be attempts by others to control or end their behaviour, and to tie it into mental health issues and get the person mental help.
Taking of harmful substances is usually referred to as 'problematic drug use' if the person is a man and chooses substances associated with men (heroin, cocaine, household DIY products like glue). It is commonly seen by the public as 'risk taking' or 'addiction' and by professionals as a 'coping mechanism.' There is much effort put in to viewing this as a 'choice,' a matter of personal freedom and money is put into providing safety advice around continued drug use, safer drug replacements and needle exchanges for those that inject. People with a dual diagnosis of problematic drug use and mental health problems often fall into a gap with a paucity of provision for people with both.
I would suggest this is a problem that exists across all gendered behavioural issues. If a women does something that is dangerous to themselves or others, it is seen as a cry for help or an inability to manage their lives. People, including the state, feel they can and should intervene in a paternal way with that person. If men does something that is dangerous to themselves or others, it is often seen as masculine risk taking or assertive behaviour that other people should not intervene with as it is a matter of personal freedom, until the point they die or end up in prison (90% of prisoners have a mental health issue).
I do not see feminism as being the cause of allowing men greater freedom to engage in behaviour or ways of thinking that increase their likelihood of destroying their lives or increase their risk of death from mental health issues. Quite the opposite in fact. But it would require a massive change in how we view the autonomy of men and women and the the problems of hyper-masculinity and femininity to change that situation, not just some tinkering with relative proportions of mental health funding given to issues that have an impact on men and women.
I do not mean this to be a polemic. I am sure there are many other factors at play.