I am not sure if triptans can use rebounds but they definitely can cause Medication Overuse Headache (MOH). This is a different headache and IHC has a proper classification for it:
8.2.2 Triptan-overuse headache
Headache occurring on 15 or more days/month in a patient with a pre-existing primary headache and developing as a consequence of regular use of one or more triptans on 10 or more days/month for more than 3 months. It usually, but not invariably, resolves after the overuse is stopped.
Headache fulfilling criteria for 8.2 Medication-overuse headache
Regular intake of one or more triptans1, in any formulation, on ≥10 days/month for >3 months.
Now its a bit a choice here between the pest and cholera. Or you take your cluster attacks as they come without MOH risk or the clusters are so painful that MOH is a small price to pay for getting rid of them.
Sumatriptan injections are suggested here. This is indeed more effective against CH attacks and its presented as the standard treatment but also keep in mind that the injections put more pressure on the heart than the pills. When I have a CH flare-up, the attacks follow each other up in rapid succession with continuous background pain in between. So for me the pills are the safer option than injections. I do not use injections but a nasal spray in the worst case (Zolmitriptan for both the pills and nasal). I can also tell you that, if you decide to use the pills, there are much better triptans available than Sumatriptan. You have fast acting ones such as Eletriptan and Rizatriptan or slow acting ones like Frovatriptan or Naratriptan.
Having said that, keep trying with the oxygen. There are several masks, breathing techiques possible you can try as Chris pointed out but if you can get it working, its a huge benefit !
For me oxygen works like magic. Rebreather, non-rebreather, high-flow, lower-flow, it all does not matter. Whenever I breath in that stuff, the pain goes away within 10 minutes in most of the cases. Even if I stop oxygen after 15 minutes with still 20 % of the pain remaining, that pain also goes away during the next minutes. With oxygen, I have managed to cut down my triptan usage with around 70 %. So it's worth further trying it but patience and creativity is key here.
All the best !