Sorry its taken all day to get a proper chance to respond, @Eden1919 . But I have reread it earlier today when I had a half-moment, so that I could keep thinking about it whilst driving, etc.
I really like what @Ainjoule said about boundaries. Where exactly you set them is up to you (like you note that you strongly encourage them to seek emergency support rather than call the ambulance yourself). Boundaries are very important in any relationship, but even more so when someone is inclined to not take responsibility for their own wellbeing/actions/etc.
Frustration about your friend's (in)action sounds quite a reasonable response for you. It can be really disappointing to see people we love making poor choices and/or giving up. Those feelings often indicate love and care, that you don't want to just give up and walk away from that friendship.
Having said that, it is so important to also put your wellbeing first. And boundaries help with that. You are not responsible for your friend's actions or mood. It is always OK for you to not answer their call or message, or to say you are not available. This is something I always stress to my friends who offer to be a support when I am in need - and if I don't think they will say "no" when they need to, then I won't reach out to them as a support (because protecting safe boundaries is so important to health and friendship for me).
Your friend may or may not be used to people setting boundaries. If they are not used to it, they may react poorly. That is not your fault. That is not your responsibility. Sometimes it takes for a valued loved one to stand firm and say "I care for you. I love you. And I am here when I can be, and to help you seek support when you decide you want to." You could mention things you are willing to help with - like driving them to a psychologist/doctor appointment, or driving them to the ED if they need to prevent SH or suicidal actions. Or perhaps to have a coffee with them once every x-often when they are not in a crisis state.
From the other side of the picture, your friends side, I can also empathise with what may be a loss of hope that things will get better - or stay "good". And perhaps a lack of purpose or meaning in life (something to live for). They may also not see SH as a "problem"; they may see it as a way of distracting or coping with intense distress in the short-term.
How to help them with that depends on their thoughts/situation. I wonder if asking them (when not in crisis) what they enjoy most is, or what they hope for the future - then you could remind them of these things when/if they are feeling a lack of meaning or purpose.
The thing that led to me stopping SH as a teenager was when my friend told me she was worried and was having nightmares that my SH would be dangerous. Knowing how much it scared them and caused them nightmares, I agreed to not SH anymore -- however I am very high on the empathy scale, so I dont expect this would work with most people! But perhaps expressing your concerns about the SH and your worries could help them see that you care for their wellbeing?
What do you wish you could tell them "if you were honest"?
And how do you think/imagine they would respond?
Im wondering if there is a way these concepts can be framed to increase the likelihood of them being received within the intended care and concern you have for them.
But regardless of their reactions, you do need to protect your own wellbeing. And asking for support here is a great step to that. 😊