As the CEO of a small company, I take the position that if someone asks (nicely) and is trying to save me money and also gets to fly (and maybe upgrade) on their preferred carrier I always let them. If they approach me in a manner that is self-serving or disrespectful I do not support a change.

My guess is that the person that you are dealing with at the consulting firm may have no idea about travel (unless they are an FTer), and my advice would be to happily use the itinerary that they provided.