What cost function and penalty are suitable for imbalanced datasets?
If you have an imbalanced dataset you usually want to make it balanced to begin with, since that will artificially affect your scores.
Now, you want to be measuring precision and recall, since those can capture a bit better the imbalanced dataset biases.
L1 or L2 won't perform particularly better in a balanced or unbalanced dataset, what you want to do is call elastic nets (which is a combination of the two) and do cross validation over the coefficients of each of the regularizers.
Also, doing grid search is very odd, you are better using just cross validation and see what parameters work better.
They even have ElasticNetCV, which does that part for you