Solved my own question above. Adding the resolution here for anyone else who may run into it:
I had to go back to the 1099-R form and change my answer to this question: Do any of these situations apply to you?
I had previously selected none of the above. Instead, I picked "I took out this money due to a qualified disaster". Then, when completing Form 5329, Turbo Tax allowed me to pick my "disaster" from a list of options that included everything the IRS considers a qualifying event.
