I have sighthounds and most of the tack I make is for other sighthound owners. I mention this because we tend to use martingale style collars so I have made a few in several different designs. The one thing that I am a bit confused by in your question is the comment about the part going under the dogs chin, the martingales I've made go around the dogs neck. Do you have a picture or drawing of the design you are working on?
One way to attach a 1/4" wide strip to a ring would be by using a slit braid, two or three slits worth would work and add a nice design elliment. If both ends need to be attached to the same ring this wont work.