Just bought a used KingBass MkII 5-string, which I'm really enjoying.

It seems to have one issue, though - the action is quite good, however I’d like to get it a bit lower, but even with the truss rod fully slackened off, there’s still quite a bit of relief in the neck. Is there any other way to get the neck slightly flatter? If I lower the action, it buzzes quite badly above the 12th fret, so I was thinking about getting the frets leveled to compensate. Is that likely to help?

Any advice gratefully received.

Should have a two-way truss rod, it can bend both ways! Maybe slightly tighten it the other way?

Thanks, I wasn't aware of that. I'll look into it and report back.