If you must follow the requirements, then the first option would be better for the simple reason that you have more data for the minority class, so you should be able to capture more of the variability in this class. As opposed to the second option where you would first have to (find a way to) select a small subset of all the data (and throw the other data out?, this seems unnecessarily wasteful).
Edit: some more clarification. The first option would see you select a sample of your data (although much bigger than the second option) and the perform CV with folds, ensuring that each fold is trained with the same ratio of positives and negatives, as your original sample. In this way you would preserve the required "natural rate" while at the same time having more data available, so I don't see how this is a worse option than number 2.
I would, however, suggest a third option, one that would require no removing of data. Keep all your data as is, but then perform a generalization of CV, whereby you train your desired model multiple times with subsamples of your data, while keeping the ratio intact (stratified sampling). So each iteration you select a new subsample, getting new data into your training sample. This way, given enough repetitions, you should get most of your data through your model, and hence get a much better look at the performance of the model. The best cutoff for AUC would then be selected from the folds, where the natural rate was in effect.
This is similar to repeated CV, where you repeat CV multiple times, each time with new folds, so as to try and randomize the training and test data as much as possible, and get a more accurate result.