There isn't really a standard for compatibility between different manufacturers of these rings and bases. A slight difference in dovetail dimensions can result in a loose ring or a badly galled dovetail. I speak from personal experience.
To be safe, I would replace the base with the proper Burris part - they're not very expensive. You can try installing the Burris ring in the Redfield base. If it turns easily, its probably too loose. If it resists starting to turn, don't force it because you could damage the ring, which isn't cheap.
Otherwise, I like Signature rings. However, I've found they can loosen up during the first 10 minutes after torquing the ring screws. I always come back after 10 min and re-torque.