I'm not familiar with the chicco nextfit. My radian will fit although it can be tricky to install rear facing (i think we are likely going to forward face it on the plane). Delta does not have airbags in belts here, so you're good in that respect.
How old is your daughter? Do you have any other carseats?
Delta does publish a helpful seatfit guide (there are at least two versions of the 777 so make sure you have the right one) - here's an example:
http://www.delta.com/content/www/en_...200er-777.html
I would assume the narrowest measurement is the bulkhead seat width.