Cruising at FL120 - there are a handful of aircraft with oxygen systems and the performance on the UK register.
Most typical 4 seater 160HP variants will get you up there no problem (PA28's / 172's etc....).
It is worth mentioning that not all aeroplanes can be put on a permit, for example you couldn't buy a 172 and stick it on a permit
AFAIK. If Cessna ceased production and support then you probably could......