I only know of two GHS inspectors, and one, Gustavus H. Scott, can be ruled out as he inspected firearms from 1828 to 1873, according to one source. He was a Navy Officer.
Gilbert H. Stewart, Colonel, U.S. Army, served as an inspector from 1915-1919 and then again from 1938-1940, although he is shown in one source as inspecting Colt and Smith and Wesson Revolvers, and Colt M1911 and M1911A1 pistols. It could have been him, time wise, at least for the conversion of the gauge to .30-06.
Still, it may not be either of them. The 1907 dates on those gauges would seem to rule out Col. Stewart, and Commander Scott seemed to inspect for a very long time, 45 years according to my source which is Steve Fuller’s book MARS TM-157 dated 20 April 1982, “United States Martial and Collectors Arms,” which deals primarily with inspection marks and the like. I do not see a man as an active inspector from 1828 until 1907, which I feel is patently impossible.
I just throw this out as perhaps a beginning for further research into the initials GHS.