Chile or Argentina for my 40th Birthday

I’m turning 40 this year (yey!). I’m thinking of a once-in-lifetime trip to either Chile or Argentina. Can you help me decide what country to choose based on the information below:
Schedule: sometime between August-September
Days: 9 days (including flights)
Airport: NYC (JFK, LGA, or EWR)
About us: family of 3 (with 1 year old baby)
Must see: Patagonia (either Chile or Argentina side)

My birthday falls on the 1st week of September but I am open to traveling other months as well if that’s better for the view or the weather.
Since we have a baby, camping is out of the question but we are open to a light trekking to see a better view of the mountains.
Any advice would be appreciated!