Silicone toys won't melt together. Even different types/grades of silicone from different companies. They also won't melt on glass or hard plastic. It's PVC and jelly that to the melting.
If you want to trust a cock ring to keep your toy from going in too far, that's up to you. I wouldn't do it, personally. There are lots of readily-available anal-safe toys out there. (Heck, if you really like Liv, get Billy. Same size, same motor, but with a lip at the base to make it anal-safe.) I'd recommend using a ring that's got a flattened inside, like Doc Johnson's Platinum Cure silicone rings, to make the ring less likely to roll on your toy.
BUT I can tell you that the ring you've mentioned will not damage Liv. I've used the Island Rings, which are TPR silicone and from the same manufacturer, on my silicone dildos (to strap them to a pillow) and nothing melted.