Sulky Rayon is fine, super strong, I have used it LOTS and lots! It's fine on your bed quilts AND your runner.
The runner is a great place for you to experiment though, you might decide you don't like the way it wears. I believe they used to say avoid Rayon because it wasn't colorfast...but that's not really true these days. I say, you're fine. If you love it, use it.
Edited to add - if you have problem with it breaking it doesn't have anything to do with the cotton, it's likely the needle. Use a nice fresh needle, I use 70/10 microtex sharps for Rayon thread, and usually a lower tension.