‘Gotten,’ in contrast, often implies the process of getting hold of something:
What choices have I gotten to make? You are emphasizing the process of getting something rather that the final outcome - the possession of that thing. Here the thing is choices. You are emphasizing that, in the past, you have had the opportunity to make choices and reflecting on which ones. You were able to choose.
What choices have I had to make? You are also emphasizing the fact that you were obligated to make choices in the past. Again, you are reflecting on what choices these were, but the emphasis is on the fact that you had to make these choices (were forced or obligated to do so).