Maximising prediction accuracy of the majority class in an imbalanced dataset
You are thinking about this the wrong way. If all you cared about was the majority class, you could just predict everything as belonging to the majority class. You'd get 100% of them right. You would have lots of false positives, but you don't care about those right? 
Ah, if you do care about the false positives, then that means you actually care about the minority class after all. The more things in the minority class you predict correctly, the less false positives you have. 
This is two sides of the same coin.
When talking about imbalanced datasets, most articles would refer to maximising the prediction of the minority class (e.g. for fraud detection). I have an imbalanced dataset (ratio approximately 1:20). where I am interested to achieve the highest prediction accuracy for the majority class. My work is in Python. Possible solutions I have researched are:

Oversampling of the minority class
Changing the loss/cost matrix for some classification models

What are the pros and cons of using each method? Are there any other methods I could try?