Some things to think about....
The old maps show Cochran's Academy in the Choctaw Nation on the Ft Smith-Ft Arbuckle Road. This fits with the "front door to the Chickasaw Nation". The Chronicles of Oklahoma, as I recall, place it on the south side of the Boggy from the little community of Frisco. You might also call the latter location the 'front door'. Have you found any other descriptions/documents that would place Cochran's Academy in the Choctaw Nation? Is that the correct location for the residence of Robert Cochran instead of near present Frisco?
You stated in an earlier post that the "Rock Academy" may refer to Cochran's Academy. I supposed it to be Wapanucka Academy where I believe a number of Creek refugees were camped. Still thinking about this one... are their any clues that point to one more than the other?
I absolutely agree that the majority of the time was spent north of the Canadian. It seems Phillips wanted to brag about his military exploits but clearly a critical part of the expedition was foraging between the Canadian and North Fork of the Canadian (now commonly called the North Canadian River). The supply situation at Ft Gibson was always marginal and he had many refugees to feed.
I'm so glad you're looking at this. A source I haven't yet been able to pursue is the Maxey Papers in Austin. Curtis Payne has copies of a portion of these but didn't find any clues. I have a portion I got from Gilcrease but they deal almost exclusively with spying around Ft Smith. There is more in Austin (TX State Archives) to be searched.