I have a group of 14 going in August. I have been told to only book LATAM or Avianca for these flights. I have several questions I am hoping someone can help me with.

1. Is one airline better than the other for the flight? Is it really best to only stick with these two?
2. Avianca is cheaper and includes a bag. When I called in for group sales, they are quoting me a higher price than I can find online. He said "that price isn't the true price". Any ideas? LATAM hasn't called me back on group sales even though it has been a week.
3. If I go with Avianca, I can book it through United. Is this advisable?
4. Should we avoid using CC points like Chase Ultimate Rewards? I have heard it is best to book directly with the airline (ties to number 3).

Our group did end up booking the Holiday Inn Lima (thread below).