Leave of Absence from PHD
Universities have procedures for that. Moreoever, universities are run by people who have lives, families, friends in such situations and who can understand and sympathize. It sounds very much as if the people in your department fall into this category and are supportive of you.

What matters is that you get your footing back and get healthy again. If that means taking a leave of absence, then that is what you should do. If there ever comes a need to explain what you did for the semester or year that you took off, you can always say "for health reasons" -- which is true, and which everyone will understand is an important and legitimate reason to take time off, and which nobody is going to count against you.

Of course, the reasons you cite against it are all very good as well. I don't think anyone here can suggest useful strategies other than finding some low-stress job that helps you pay the bills, get you back on your feet, and allow you to come back to the program in better shape and with better prospects. The alternative, however, is not all that appealing either: as you state in your post, if you keep doing what you're doing, you may fail out of the program and then you're in no better situation either. In fact, the situation may be worse since it's easier to come back after taking a year of absence than come back from being kicked out of the program.

Ultimately, what matters is that you get healthy again. Everything else should be a secondary consideration.