Reframing multilabel classification with imbalance in "both" directions
I don't think there's any obviously best option. I'd suggest trying a few reasonable solutions, evaluate on a development set and then pick the one which performs best. Don't forget to try with the original data as it is, resampling doesn't always work better.

Upsample the non-target class? Downsample the target class?

These two options are very likely to give the same result (assuming you use the same proportion).

Reframe the problem as &quot;not drama&quot; to maintain underbalance for every label?

For the classifier it would be exactly the same.