Hi Glimmer.
I've got a daugter with different problems, but which started showing at a similar age.
There's a few hard "facts" that will get in your way.
You cannot legally compel her to receive any treatment without an involuntary admission. If you trick her into going to the hospital, as well as losing her trust, the moment she makes it clear that she's there against her will, the psych/medical staff can't legally touch her. At 16, they can't even talk with you about her treatment without her permission. This is a regulation that makes reasonable sense when it's in relation to physical care, but is at times very obstructive when it comes to psychological care for someone who lacks insight or is obstructive to their own care... or whose families simply need feedback about what they're dealing with.
Psych specialists' attitudes to involuntary treatment orders vary from one place to another, but generally they're reluctant to put them in place unless the person is a real danger to themselves or others. Even then sometimes they won't. The problem with an involuntary order is that it can make things worse. The majority of mental illnesses are exacerbated by feeling out of control. It's the basic precursor for most forms of depression. With persecutory delusions in the mix.... yeah, it would just confirm what she already thinks is happening to her. 😞
So that's the bad news side. "Best advice" side, do everything you can within reason to keep her trust. In doing so, be as honest as you think you safely can be. If she sniffs out any lies or half truths, you're immediately one of the persecutors. Talk with her about where she feels safe, and who she feels safe with. If there are any medical professionals or people in counselling roles that she trusts, see if they're willing to try to talk her around to trusting the treating psych.
Sorry I can't give you much of a way forward. Hang tight, get help and backup for yourself, and keep loving her. Sometimes that's the thing that proves to be the lifeline.