How to make a classification problem into a regression problem?
So, the direct answer here is clearly NO. 

The answer comes from the definitions of classification and regression. In a classification task what a model predicts is the probability of an instance to belong to a class (e.g. 'image with clouds' vs 'image without clouds' ), in regression you are trying to predict continuous values (e.g. the level of 'cloudiness' of an image). 

Sometimes you can turn a regression problem into a classification one. For example if I have a dataset of images with labels of their cloudiness level from 0 to 5 I could define a threshold, i.e. 2.5 and use it to turn the continuous values into discrete one, and use those discrete values as classes (cloudiness level &lt; 2.5 equal image without clouds) but the opposite is definitely not possible. 

Here's also a link to a similar question Could I turn a classification problem into regression problem by encoding the classes?

To solve the problem of imbalanced classes there are many ways, not just oversampling, you can generate artificial data, add class weights in the loss function, use active learning to gather new data, use models that returns uncertainty score for each prediction (like bayesian networks), I'm sure here is plenty of answers and strategy you can try.