Dealing with correlated features when calculating permutation importance
Using mutual information namely the correlation of each feature and the output is not something that can be helpful very much. The reason is that the correlation coefficient is just capable of finding whether they are linearly dependent or not. Other than Gaussian distribution, as far as I know, it is not even capable of finding out whether the inputs are independent or not which means if you see the coefficient is equal to zero, if you don't know the distribution you cannot conclude that they are independent but you can say they are not linearly dependent. In real-world applications, it scarcely happens that features just have linear dependencies with the output. Consequently, approaches like this are not very helpful because you may have two features that can have a better relationship with the output but none of them has a good separation with the output.

For cases that you have a lot of features, you can decide to choose other feature selection and extraction methods. I guess for your case wrapper method may suit better. In the wrapper method, you don't have any criterion and you just search through all possibilities of features using heuristic-based methods in order to find the best sub-features which have the best cross-validation accuracy.