I don't know what rayon machine embroidery thread is, so am not sure if it would work. Does it have some cushiness or loft to it like Wooly Nylon? Or is it very smooth and slick? One thing to try: take a short length of your yarn between the fingers of one hand and kind of saw at it with the thread you're considering using for reinforcement (kind of like playing a violin). If the yarn does not abrade or weaken, I think you could use it.
You might want to wait till someone else chimes in here...I've got lots of sock and yarn experience but zero rayon thread experience!