Model-agnostic variable importance metric
Have you looked at the permutation importance approach in the eli5 package?

The idea is that instead of retraining the model without the feature, which is computationally expensive, they replace each feature in turn with random noise in the test set. To get random noise that is drawn from the same distribution as the original feature, they just randomly shuffle that feature.

Note that, as with feature importance in decision trees, this measure is biased against categorical variables with low cardinality.