I wish I had the right answers for your situation.. I can see how hard everything is for you dealing with your own mental health, and all the responsibilities you have on your own.
You mentioned you have doubts on her diagnosis.. Is this because she had only verbally said it to you, without any sort of evidence from a doctor?
Sticking to tough love is exhausting. I think maybe the most important thing to remember is actions speak volumes but words don't.. I.e say if you give her a deadline to have a job by x month and she doesn't, you might find yourself backed in a corner.. Asking her to do something vs. not doing it for her so that she has to do it.
If she stays in her room all day not doing anything.. Does she cook/clean after herself, do laundry, have any chores or responsibilities as a member of the house like that? Or is she entirely dependent on you, also financially?
If she says she wants to be independent, don't pick up after her. Treat her like the adult she is, and I think this can be done without seeming rude or mean by any means. End of the day she is responsible for herself if she has been diagnosed with a mental illness and is refusing to get treatment or live her life. It's not a free ticket to cling on to you, or an expectation of you as her parent by any means. I think you can be a great support system for her by not giving her the option of doing absolutely nothing.. Ie. maybe starting small like making her cook for herself, telling her to get groceries.
I understand this is much easier said than done and I'm not a parent.. But I always felt guilt and like a disappointment if I didn't pick up after myself and do what was expected of me. I learned it shouldn't be a matter of them asking me to do something, it's telling me to do it.
You are not a bad person for having the expectation of her to live her life, pull her weight.. That's not being dismissive of her mental health at all. I think if anything her thinking that doing nothing all the time is allowing herself to dismiss her ability to live life because her mental health is in the way.. She's making the choice to use it as an excuse.
Give her the space to talk about what she's going through, what it is that's difficult for her, be empathetic to that. Normal tasks are going to be hard for her, but never impossible. The longer she avoids what she needs to, the more impossible I think they'll seem.
I hope you've found some parts of this helpful in some way.