You could ask her when would she break confidentiality and not - and then you can have some reassurance from her directly.
If you can commit to what is called a safety contract, basically a promise you won't try to end your life without asking someone for help, then she is extremely unlikely to try to hospitalize you.
The only time one she can share info with your doctor without your signed written consent in the US is if your life is in imminent danger of ending, (or if someone is imminetly homicidial or a minor is being actively abused.) Outside of that, she legally can't tell your doc anything unless and until you are ready.
If you have signed a release form for her to communicate with your doctor, then you can cancel it.
That all being said, if/when you are ready, it might help for her and your doctor to communicate so they can best help you get through this and find the safe relief from the pain you are in. But it's not required.
My therapist and I discussed my attempt in detail, and it was hard, but she was quite unphased about it. It really helped.
In a nutshell, it is safe.