Hello!
I assume the therapist you see is not through your school but rather an outside provider?
The therapist I see is through my university counseling services...
If seeing your therapist sooner is not an option, maybe it is worth it to see if your campus counselor has an early availability (sometimes they are crazy busy).

As far as disclosing to someone university-related:
In general counseling/health records will be kept confidential if no other actions need to be taken. For example, if you need medical leave or are sent to the hospital that would be different bc the university would need to approve of that and you coming back.

I have had similar fears, not about getting kicked out of school, but of getting sent to the hospital. A good way to handle it is to ask right away. So you can go in and ask the counselor what their policy is if a student is doing this or thinking that. If you phrase it as a hypothetical question rather than a personal admission you can gage what their reaction will be and any potential consequences.
When I switched counselors I asked her what her threshold was for sending people to the hospital, like does she do it often and why.

The only other problem could be that this counselor would not have any background. It would depend on you and what you were struggling with. I have difficulty opening up to people and telling them what my past/current struggles are so I don't think I would benefit much from one session to get support. But again, if you feel this is worthwhile and what you need to get the help you need it is totally worth a try!

Like the previous poster suggested maybe try reaching out to your current T if that is an option. If not, it may be beneficial to go talk to someone if you feel that would help you feel better.

Take care darling.