There is a feature selection independent of the model choice for structured data, it is called Permutation Importance. It is well explained here and elsewhere.
You should have a look at it. It is currently being implemented in sklearn.
There is no current implementation for MLP, but one could be easily done with something like this (from the article):
def permutation_importances(rf, X_train, y_train, metric):
baseline = metric(rf, X_train, y_train)
imp = []
for col in X_train.columns:
save = X_train[col].copy()
X_train[col] = np.random.permutation(X_train[col])
m = metric(rf, X_train, y_train)
X_train[col] = save
imp.append(baseline - m)
return np.array(imp)
Note that here the training set is used for computing the feature importances, but you could choose to use the test set, as discussed here.