Once you create the double helix elipse you can delete history and then the issue of double translation will no longer exist. Then you can delete or save all the construction curves, combine just the double helix surfaces, and animate as you wish. But if the other method is working for you then that is cool too.
Well I'm still thinking atm, trying to see which way would work and what / how it will look in the end + the animation effects must be considered, as I want the circles (not as elliptic as seen in the picture) to rotate like a disc but also revolve around that sphere in the center. If I'd make them elliptic, they'd be hard to rotate like a disc properly (unless fluids or something are used, or there's some other trick to make them move while following a precise trajectory that changes the chain shape at the same time).