It doesn't look like you're going to get much of a response here, Florin. Have you tried the various Facebook groups? They have a lot more participating members. The group called "Cluster headaches (trigeminal autonomic cephalalgia)" is highly regarded here, but there are also others.
I'm perplexed when you say "oxygen for the migraine," because oxygen is never, or almost never, prescribed for migraine. It just isn't very effective, or effective at all. Only for cluster headache.
In a city where practically every major medical center has a headache clinic, I'd be surprised if you couldn't get an appointment, and if your condition is cluster headache, it would be surprising if a big-city, hospital-clinic physician or PA didn't prescribe oxygen. Montefiore, Mt. Sinai, Cornell, NYU, Columbia. Maybe you've tried all these, or maybe you are hoping for a first-hand recommendation.
As Bosco' said, you're not really sacrificing anything to go with welding O2 .... Over the long run, it could even prove less costly than a medical prescription, depending on what your insurance covers.