I don't really know what you are trying to accomplish but from the description, I sure hope that sprocket has a bearing in it.
You really should get a micrometer and a piece of stock (like drill rod or a finely finished piece of square stock, etc) and close in on the stock and back off some known amount, and then slide and jiggle the stock to get a feel for different fits. Then you would know things like just how loose a clearance of 0.005 feels right up against the fit, and just how loose it feels when when amplified by a length of 4" away. It's not good to blindly design with numbers with no physical experience.
I am assuming that if I am mounting a component that is 1.25" wide, the distance between the grooves should be exactly 1.25" as seen below:
Well, no. Because your sprocket may not be exactly 1.25". You need to know the thickness tolerance of your sprocket and how tight a fit you want/need.
the ring will be under slight tension once installed,
I don't know about aerospace requirements, but from my limited experience cutting grooves for retaining rings for regular stuff it doesn't seem critical. The ring doesn't seem to really need to be under tension in the radial direction. I recall many retaining rings for motors being quite loose on the shaft. I'm not actually sure whether the preferred scenario is to be under tension or to not be under tension. But for regular stuff, it seems that all you need is for t groove to not be so deep (i.e. the groove diameter to be so small) that the ring can sink deep enough to let something slide off. and for the width to not be so large that the ring can excessively tilt in the groove which would allow you to twist the ring off. They seem pretty idiot proof to me.
Look on McMaster Carr for retaining rings. They have recommended groove tolerances for different retaining rings you can draw from as well as the tolerances of the ring and grooves.