It probably makes sense to go through each of Barker's projects and note which ones (a) have a record of Travis involvement, (b) which ones would make sense for Travis to be involved, and (c) which ones were most likely Barker on his own. I'll try to do this chronologically the best I can.
Garden City GC -
Certainly Barker's first work in the U.S. Prior to Barker's arrival in the fall of 1907, the wheels had already been put in motion to further improve Garden City. While Travis was undoubtedly the driving force behind the changes, Devereux Emmet was still serving on the Greens Committee and he and others would have had a say and perhaps a good bit of input into the proposed plans.
Oct. 14, 1906 Boston Globe -
There are some interesting questions that arise in contemplating the hiring of Barker. Who exactly was responsible for bringing him in? Was this meant to be a long term engagement? Did the club have any inkling of Barkers links course knowledge when they hired him?[
In any case, Barker and Travis seemingly worked side by side on the course. Both men were playing a lot of golf between 1907 and early 1911, and with Barker's other obligations as pro this work would not have been measured as a full time job.
Feb. 1908 Golfers Magazine -
Aug. 1909 Golf Magazine -
Jan. 7, 1911 Brooklyn Times-Union -
Barker would leave GCGC for Rumson in the Spring of 1911, drawing an end to the working relationship here.