Okay, so my main issue currently is the intro to my sci fi military story that I've pretty much fleshed out the most major key points, maybe four or so at the most, that I'd really like to keep. But I have a character that I'm having a bit of trouble placing in the timeline for plot's sake. Originally this character (we'll just call him private Joe) was going to be the catalyst for my villain to become her fulfilling role. He was supposed to be intentionally betrayed by said villain (we'll call her private hothead) and then my main character was going to take a dive deep into the realms of depression and heroic BSODs (for those of you who don't know, BSODs or "blue screens of death" refers to the thousand yard stare most patients that suffer from PTSD experience.) After some thought I believed it would be better for my main character's sake to begin her fall into madness at the beginning so that her squad mates had a chance to reinitialize the person she used to be. But now I can't decide between the two. So, for sake of plot and placement, should a supporting character's death be closer to the beginning or during the climax?
I'd say it depends on your pacing and what kind of precedent you want to set. Killing off a supporting character towards the beginning could send the message that anyone could die at any time and raise the stakes. Would that idea support your theme, maybe add in a new layer that you might want to explore more? Killing him off later after having had no casualties, though, and having had more time for the reader to get attached to him, would be more of a shock. Which would play more into your mc's ensuing BSOD - the reader can better relate to those feelings having also lost this character. You do want to make sure there's a realistic sort of timeframe for your mc to get through the BSOD but that really depends on the character. Maybe she doesn't need that much time. I'd also consider the nature of your climax. Without knowing anything about I don't know if this is applicable, but having whatever else is going on in addition to the betrayal and death of Private Joe could make things seem a bit scattered and unfocused, so spacing out these two seemingly pretty big plot points might be the better idea just for streamlining purposes. But again since I don't know if this is likely to be an issue with your climax (the two events could be tied in together really well), use your own judgement.
What kind of state do you want the main character to be in at the final battle? It sounds like you're leaning towards having her be a relatively stable person by that point, in which case the death should be earlier. However, it's also possible to have the final battle be the point where she takes the first steps towards recovery and is able to defend herself against the villain, with the implication that she still has a long journey back to health. If you want to do that, you can have the death be later.
How much does the supporting character have to contribute? In addition to the points raised @Feo Takahari and @izzybot you should consider how important is the content that would be lost in killing them early. And if there any planned scenes that you would want to keep, is there any character you think could replace this character in those scenes? If there are any important content concerns I would consider keep the character till the middle.It might just be worth it for the character's role, especially in establishing the connection with the main character that makes the death tragic for the audience.