I was thinking. Would it be a good approach to check your features one by one (assuming you have a manageable amount of them) and see the relationship they have with your target variable, if they have a non linear relationship then transform each of those features using their appropriate function for each case to make them linear? In my mind if you do this your are guaranteed to have a better Linear model and also you are able to perform hypothesis testing on each feature to see the relevance of them, giving you the chance to perform some feature selection as well.
I know that the interpretability of model will be thrown out of the window, but the model will give a much better performance. Basically you could potentially end up with a model with only engineer features (assuming that all of them have a non linear relationship)
Would this approach be acceptable and it is worth exploring?