If you were bleeding would you insist on only letting your reg doc patch you up? I hope that doesn't come across wrong but what I mean is that this is an emergency even if you aren't planing on acting on suicidal thoughts. As long as you are up front with the counselors on campus and tell them you just need a support until your appointment I think that's what they are there for. To support the students.

If you have an emergency number or email for your reg therapist call. When you see your reg therapist maybe discuss what you should do in a crisis and develop a plan for next time. Even if you never use it again it's nice to know that there's a plan.