If you have significant pain all day, and the attacks that seem like CH are "bursts" as you write (which I think of as briefer than a typical CH attack), and if a triptan doesn't help, it seems possible to me that you might have one of the CH "lookalikes," some form of hemicrania, maybe paroxysmal hemicrania (Paroxysmal Hemicrania: Causes, Symptoms & Treatment) or hemicrania continua (Hemicrania Continua: Causes, Symptoms & Treatment). I'm just guessing here: you can look to see how well the symptoms match. Unlike CH, hemicrania is generally not responsive to triptans or oxygen (or busting, except for some brief relief of maybe a day or two). So one test would be to take the medicine that treats hemicrania, Indomethacin. Another would be to try oxygen.
As I say, I'm just guessing here. Sometimes people "try" Indomethacin, but the dosage isn't enough, or it's not continued long enough, to get a real test. This is the recommendation from the International Headache Society: "In an adult, oral indomethacin should be used initially in a dose of at least 150 mg daily and increased if necessary up to 225 mg daily. The dose by injection is 100-200 mg. Smaller maintenance doses are often employed."