Not sure that suicide (apart from the moral connotations we include as part of its definition) would be much of a distinctive. Male black widow spiders don’t appear to be deterred by the thought when they’re planning their daily itinerary. But you probably want something a little closer - at least in the mamallian class? Just a quick glance at a site on the web seems to indicate that what appears to be animal suicide (among dogs or bears or cats) while not common - is not unknown either. (By the way - did you know that some mother spiders allow their young to literally eat them? I did not know that.) Of course in all this, one can argue that true suicide has to be with planning, knowledge of mortality, moral implication etc. So one can easily just carefully define it until the only suicides that really qualify as such must only be human. If that’s what you want to do, then the true distinction you are after is farther up the chain, and not all the way down at suicide.
If you are looking for more background on the topic of human distinction then the newly posted podcast about it (that you’ve already listened to I believe) may take us farther.