Reframing multilabel classification with imbalance in "both" directions

Consider the multilabel problem when asking "does the sample belong to this class" with, for example, a movie label dataset where almost every movie is labelled "drama" because of this label’s vagueness. It also has a few rare labels such as "French late 60s noir".

To both boost recall, as well as maintain compatibility with most classifiers, we can both resample and train a binary classifier for each label individually (drama: yes/no, noir: yes/no).

But while "French late 60s noir" might be a 0.05:0.95 imbalance that requires serious effort for the model to guess, the "drama" category is so widely applied that the problem is reversed, i.e. 0.95:0.05.

How should one logically best proceed to maintain recall over precision, now that the "target class" is 95% of the labels (kind of breaks with the definition)? Upsample the non-target class? Downsample the target class? Reframe the problem as "not drama" to maintain underbalance for every label?