But it’d still have to change length by about 6 km every 3 Earth days.

Maybe this is crazy, but make it a giant jumprope. When they're closer together, the jumprope makes a tighter circle and spins faster. When they're further apart, the jumprope makes a larger circle. Put a weight in the middle so it has enough centripetal force to keep the rope taut.

Trust me, it works, I thought about it for two minutes.