how to select the metric to optimize in sklearn's fit function?
This is not possible with Support Vector Machines, as far as I know. With other models you might either change the loss that is optimized, or change the classification threshold on the predicted probability. 
SVMs however minimize the hinge loss, and they do not model the probability of classes but rather their separating hyperplane, so there is not much room for manual adjustements.
If you need to focus on Sensitivity or Specificity, use a different model that allows maximizing that function directly, or that allows predicting the class probabilities (thinking Logistic Regressions, Tree based methods, for example)
When using tensorflow to train a neural network I can set the loss function arbitrarily. Is there a way to do the same in sklearn when training a SVM? Let's say I want my classifier to only optimize sensitivity (regardless of the sense of it), how would I do that?