I think if you want to keep it from seeming edgy, you have to make the laughter not the whole of what's going on, and show other signs of the breakdown such as tears, and what not. Even if you're fronting being happy, if you are having an actual mental breakdown, it's going to show through. It is definitely possibly for it to not come off edgy, but again, it's about the laughter not being the whole of what's going on. in a real mental breakdown the facade is going to crack, and probably fall to pieces. you can start with him laughing but have those laughs slowly turn into something more like gasps, and then choking sobs as he tries to keep smiling while seriously losing it.
I have something a bit like this in the last chapter of my with a character who's tried to show a happy front the whole time having a mental breakdown and still trying to keep up the front:
He doesn’t budge, staring at the blankets like his life depends on it.
The surprise and bafflement subsides into confusion and concern, as I watch the torrent of torment in his expression, his pale face changing colors from white to blood red.
“Why didn’t you say something…?”
He tries to laugh, but the sound is fragile-
*
Perhaps you'd want it to be less subtle, or possibly not. But you can definitely laugh while having a meltdown without coming across as the joker. And as you mentioned that this person is crying and laughing about their trauma, then I think the above mentioned tactics might be your best bet.
Also having them say something super depressing as they're laughing and crying at the same time will help it seem less like the joker and more like they are depressed having a meltdown.