Hi Beebs and welcome
Headache diagnosis at ER is very hit and miss as even most neurologists do not have the skills or experience required to diagnose or treat complex headache types like CH. What you really must do is to work with a headache specialist to get a confirmed diagnosis which then will lead to the right treatment for whatever it is that your partner has.
The standard diagnostic criteria for CH can be found at - Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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A) At least 5 attacks fulfilling criteria B-D
B) Severe or very severe unilateral orbital, supraorbital and/or temporal pain lasting 15-180 minutes if untreated
C) Headache is accompanied by at least one of the following:
ipsilateral conjunctival injection and/or lacrimation
ipsilateral nasal congestion and/or rhinorrhoea
ipsilateral eyelid oedema
ipsilateral forehead and facial sweating
ipsilateral miosis and/or ptosis
a sense of restlessness or agitation
D) Attacks have a frequency from one every other day to 8 per day
E) Not attributed to another disorder
At the moment, from what you've described, his headaches do not match all of these criteria, especially on the headache duration which at over 72 hours is a long, long way outside the upper limit for CH of 3 hours. CHs also start and stop very quickly, with the transition from being pain free to in pain / pain to pain free being seconds to a minute or two (typically), not the slow build up that you described.
Whilst there has only been one, not five, this isn't as significant yet.
What makes CH so hard to diagnose is criteria E, where ALL other possible causes must be ruled out first. There are tens of other possible causes for the CH symptoms that will need to be ruled out as they give either the exact same or very similar symptoms. ER isn't normally the place where you get all the required tests done.
Did they give him an MRI or CT scan? This is normally given to rule out things like brain tumours.
Another reason to question their level of CH knowledge is that pain killers are not a standard CH treatment as even the strongest narcotic pain killers do not really touch the pain of a CH, but instead rapidly result in issues like dependency. If you search through the forums here you'll find too many examples of where pain killers have been given only to cause issues (it happened to me when I was given codeine for my CH before they diagnosed CH).
For using oxygen for CH, this should have been at a high flow rate (15lpm or higher) using a non-rebreather mask (this has a bag which fills up, not nasal tubes or a mask with holes in). Lots of oxygen info is at Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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It could still be CH, but you really need to ensure that it is the right diagnosis.
This is a scary time, it's one we've all gone through. But getting the correct diagnosis is essential to finding the right treatment for whatever it is.
Start to keep a headache diary which details every headache with the time, date, duration, pain level (1-10), symptoms and possible triggers. This can help with the diagnosis.
If it is CH, a very common trigger for most people is alcohol, often bringing on a CH in minutes. So it's probably best if he avoids it.
Where are you in the world? If you post say the state / country, then it's possible that someone here can recommend someone from personal experience.
Also read up all you can about headaches, here and elsewhere. Knowledge is critical for dealing with them (I've also sent you a link to a lot of CH info). Then ask any question you can think of as together we've a lot of experience around CH and getting diagnosed.
Finally, you're a great supporter doing this research work for your partner.