In that context, the term "equipment strap" refers to the practical function of a uniform's shoulder-straps. Notice the buttonhole on them. You would unbutton that end of the shoulder-strap, put on your over-the-shoulder swordharness, and re-button the shoulder strap. That would prevent the swordharness's strap from sliding off your shoulder.
The shoulder-straps could also hold another piece of equipment in place... the shoulder-scales, which are also known as epaulletes.
BCH and SBB, although getting the catalog's actual printing date would be best, I noticed that in the Ebay seller's photos of pages from the M.C. Lilley catalog, the page on "Alterations" says July 17, 1912.
That figures in with what I was expecting, because the "heyday" of the KoP and UR seems to have been reached in the prosperous 19-teens through the 1920s, until the arrival of the Great Depression.