It's not uncommon that a PA is more helpful than the doc. This is the original O2 study, fully consistent with medical research standards: https://jamanetwork.com/journals/jama/fullarticle/185035 It wouldn't hurt to print it out and bring it with you. There is also some more recent research, less rigorous, showing that higher flows are better. All doctors and PAs have some kind of app that gives them core information about a condition. They will all show that oxygen is the #1 abortive (usually triptans are also #1). A commonly used app is UpToDate. You can ask the PA to look up CH. An O2 prescription should read something like this (write it down and bring it with you, because a lot of med professionals don't know how to write it): "Oxygen therapy for Cluster Headache: 12-15lpm up to 15 minutes with non-rebreather mask." There are abbreviations in there when it's formally written, but that's the content.
You might also look here for a little more info about the other pharma things you might want. https://clusterbusters.org/forums/topic/6213-basic-non-busting-information/ The linked-to article under the heading Pharma is clear and helpful (and also states that O2 is the #1 abortive -- I guess maybe you'd want to print that and bring it with you, too).
So, sometime people have a CH "lookalike" that is most commonly some form of hemicrania. You can look that up -- hemicrania continua, paroxysmal hemicrania, any of them. As BOF says, oxygen is generally not effective against hemicranias. There is, however, a pharma drug, Indomethacin, that is effective. Some medical writers have said that if there's any doubt about whether a patient has CH or a hemicrania, they should do a course of Indo at the beginning of treatment. (Indo is very hard on the gut, for most people.)