I am positive there is some connection between migraines and CH, but my own experience is not particularly in favour of psychedelics helping migraines. Until my late 20s I had occasional migraines and no CH, then when CH started I got no migraines ever - so whatever happened to my neurochemistry affected both conditions.
The past 2 years or so I have been using RC seeds to treat my episodic CH with probably a 80-90% chance of them aborting a cycle, it rarely fails. However, it is very common since taking RC seeds for me to get one or two migraines after the 3rd dose and this signals an end to the CH cycle. I have started getting occasional migraines in general again after a >15 year remission.
Therefore this suggests to me that LSA (or RC seeds specifically, NOT mushrooms) definitely affects the migraine circuit of my brain but , in my case, not in a positive way.