Hi Ryan and welcome
Time to get your GP educated a bit as the South Island should work just like the North Island for how you should be able to get your CH looked after.
Whilst the brand names for the injections of Imitrex or Imigran aren't funded, we can get the generic form of the injections, i.e. sumatriptan, on a standard doctors prescription.
Asking a good pharmacist is the most reliable way to tell if something is funded. And even for stuff which isn't funded, they can normally obtain it from their wholesalers, although expect the price to be a lot higher than getting say a dozen (or more) on a NZ$5 prescription (US$3 for those in the US).
It is possible to get medical oxygen in NZ too. I've had it direct from BoC and via my local DHB (CMDHB), although it took a bit of effort to get the DHB to do so. I wrote extensively about this at - Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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For more info on using oxygen - Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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People have also used welding oxygen when they can't get hold of medical oxygen, telling the gas supplier that they are taking up welding or similar.
Having both the oxygen and the injections means that you can kill off CHs pretty quick. And both can be effective in about 5 min or less, which is so much better than using the sumatriptan pill or riding it out. Plus you can take the injections where it isn't as practical / possible to carry around an O2 cylinder, like flying, etc.
There is a store on this forum where you can get a great non-rebreather mask which is perfect for CH (Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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). It is a lot better than the standard disposable ones you normally see.
Another trick is to drink an energy drink with caffeine / taurine as quickly as you can at the start of a CH (Red Bull or similar) and it'll help cut down the CH duration / intensity. This should help whilst you've only got the sumatriptan tablets.
With verapamil, whilst most people respond well to between 360mg and 480mg a day, sometimes people need more and have gone to 1000mg a day, but working with their doctor and including regular ECGs as verapamil can modify the PR interval, hence the regular monitoring. How high you can go can be dependent on your personal medical history.
There are also two forms of verapamil, which are the instant release and the sustained release. Some people find that one form works better for them than others. No easy way to work out which will work best for you other than trying both. But remember it takes about 10 days for a dose change with verapamil to become effective, so don't change too soon.
An alternative to verapamil is using vitamin D3 which has been effective for over 80% of people who have tried it. I've been CH pain free for over 5.5 years using it, so I can strongly recommend you read up about it - Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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We all know just what its like to have CHs as unless you've had them you can never really understand just what it is like to have them. So given your limited arsenal of weapons against CH, having to ride out most of them, it is no wonder you're feeling like you're having a bit of PTSD. I'd also get that looked at too as it can be addressed too.
Hope you'll be winning soon and ask any questions you like. You've people here from all over the world (even Aussie) with lots of CH experience and knowledge.