How can someone get FAA commercial flight training as part of an aviation degree?
In the US, there are 2 ways to become a certificated civilian pilot: Part 61 and Part 141. 

Part 61 is done in informal classes and flights at a local flight school. 

Part 141 is a professional pilot training program, with uniforms, classrooms, a full-time staff, and a uniform fleet of aircraft.

There are numerous US universities that offer bachelors' degrees using a part 141 program. You can find a list of such programs here.

Furthermore, many of these schools, such as Embry-Riddle, take the FAFSA, which is the  process for obtaining financial aid as for any other program.

After graduating from one of these programs, you will have obtained all the training necessary to work as a flight instructor, or as a commercial pilot under Part 119


Private Pilot Single Engine
Instrument Pilot Single and Multi-Engine
Commercial Pilot Single and Multi-Engine
Flight Instructor Single Engine
Flight Instructor Instrument
Multi-Engine Flight Instructor


You will have also earned either an Associates or Bachelors degree, usually in something like Aviation Engineering or Aviation Operations.

You will not, however, have flown enough hours to get a job flying air taxi or cargo under Part 135, which requires 500 hours to be the pilot-in-command.