If "they're all human" is all you can come up with, I think that essentially proves my point. Cultural factors are likely to be difficult to detangle from the whole mess of what's going on.
As far as abject suffering goes, I agree, that probably doesn't help the suicide rate. But, what I've been getting at is this: you keep referring to "absolute poverty," and I'm telling you I don't think "absolute poverty" is it. I think it's plain old poverty, which is relative poverty, which has to do with what one's local purchasing power is.
"Abject suffering" is relative, too. What you might call "abject suffering," someone in another country might just call "Tuesday." Conversely, someone used to living very well who ends up losing everything financially might feel extreme despair. Think 2008, or 1929.
In other words, add all those cultural factors on, and what an American might want to end it all over, a Pakistani (to pick a random example) might not. If you did a study with 10 countries, then you'd have 45 pairs to consider, and, like I said, that gets messy quickly.
This is roughly why I suspect there aren't any good studies that do what you suggest. Almost all the studies of this nature concentrate on a single country, and I suspect it's because the research itself would be difficult, but, also that it would be extremely tough to make any cross-cultural conclusions, which is what you want to do if you want to know what makes "humans" want to kill themselves.