I'm currently reviewing bus on Launchpad and the option to Triage them is greyed out for me. Is this because my Karma is not high enough (599 at time of writing) or is it because I'm not a part of the correct team (I'm working on papercuts).
No, Launchpad karma has no bearing on what you can access.
The "Triaged" and "Won't Fix" bug statuses are reserved to the role called the "bug supervisor" for each project or distribution. For Ubuntu, this is the Ubuntu Bugs team (actually, effectively, the Ubuntu Bug Control team), which you have to be part of in order to set the statuses in question.
You can join Ubuntu Bug Control once you've spent some time triaging bugs and have familiarized yourself with the various triage procedures.