I suppose your doc wanted his/her diagnosis confirmed by a neuro before prescribing any of the other standard CH treatments, such as oxygen. The timing of attacks that you have described surely sounds like CH. I assume you'll have an MRI to rule out anything else (CH can only be diagnosed from the symptoms, not from an MRI or any other medical test).
Being new to the condition, you might get some benefit from this "overview" document: https://clusterbusters.org/forums/topic/6213-basic-non-busting-information/.
As is mentioned in there, lot of people find that caffeine will stop an attack or reduce its severity -- and even though your attacks are nocturnal, most people find they can get right back to sleep even after that caffeine intake. Given the relative mildness of your attacks and your newness to CH, I wouldn't be surprised if a strong cup of coffee would be enough to do the job. Some people do energy drinks of one sort or another (discussed in the document).
As others have said, you'll find a wonderfully supportive community here, so please keep us informed and don't hesitate to ask any questions you might have.