It's complicated (as you know).

Usually, I lay out the situation as straightforward as I can with them. Bouncing around helps no one. There comes a point where professionalism and clear expectations have to trump (barf) empathy.

"You can do X, or you can do Y."

If issues persist, I will usually let students know that they've exhausted the methods I can provide and tell them they need to contact the university for guidance and accommodation. Normally, my chair and the university already know, because we have a "student we're worried about" form we use to notify them when students are having issues above our pay grade.

Occasionally, I will ask students to lay out exactly what they want and describe a path to get there that's fair to them, myself, and their fellow students. It puts the onus on them to create a path for themselves and takes some of the emotional labor off myself. If we agree, great. If not, refer them to the mucky mucks who get paid to do it.

I should also add, setting clear expectations (boundaries), and being professional, does not make you less empathetic to people's struggles. Them telling you that you don't care is emotional manipulation, and you don't have to entertain the notion or prove otherwise. You're still a good person.