These days you don't even have to be online to work on debt badges. Patrol XP does the work for you.
All you have to do is find a way to die a whole lot (like taking a tray of awaken inspirations to Monster Island), and log out when you hit the debt cap. Due to the mechanics of patrol XP, when you have debt, you work it off while logged out. Once the debt is gone, you start accumulating patrol XP. Likewise, any debt you would normally earn when defeated is first subtracted from any patrol XP you currently have accumulated.
All debt worked off by patrol XP counts toward debt badges, so if you have to die off a few extra times to get rid of patrol XP on your way to the debt cap, that's just bonus progress.