The metric expansion is not affecting the shape or H II gas cloud orbits of galaxies at increasing redshift, so newer galaxies (less-redshifted) and their host galaxy clusters have not themselves been pulled apart over the course of billions of years even as the clusters expand away from one another. Additionally, stars aren't disintegrating, various lunar ranging experiments don't show a cosmic component of the evolution of the Earth-Moon orbit, and "Brooklyn is not expanding"[1] (nor are optical fibre cables buried within it). Cosmic expansion, if considered as a sort of (frame-dependent) force, is very weak compared to real forces.
Why would the rope, if it's not ripped apart by tension and shear, or exposure to high energy ions and other radiation in the interstellar and intergalactic media, behave differently from Manhattan or the Milky Way? It might, but you'd have to write down a hypothesis in order to have a decent starting point for what "[the rope] expands just the right amount" means.
My counter-hypothesis, loosely, is that the intergalactic part of the rope (assuming it's taut) induces a perturbation on the FLRW metric that in cylindrical coordinates (where the rope forms the axis) quickly asymptotes to flat space; we can then apply junction conditions with FLRW there. The much shorter rope segments in the two galaxy clusters and host galaxies can be treated similarly, substituting a suitable metric in the Lemaître-Tolman-Bondi (LTB) family. (We already know how to do a "swiss cheese" cosmology where we embed LTB vacuoles in the expanding background of FLRW, using junction conditions). The ends are the tricky part: are they really able to keep the rope taut over cosmological times, or instead do the anchors end up colliding with each other eventually?[2] That last problem we sidestep a bit by not anchoring one end: it just winds around the electric generator while there's still rope left on the generator end. When there isn't any more rope, the unanchored end will tend to fall into the host galaxy of the anchor.
The more physical answer is that the rope, anchored or not, breaks into many pieces from a variety of causes. Uncoupled by whatever non-gravitational forces hold the rope together, the fragments of the shredded rope couple to the local metric around them. The intergalactic segments expand away from everything just like galaxy clusters do, the in-galaxy-cluster segments at either end fall inwards, perhaps ultimately landing on the anchor points.
Even more physical answers cast doubt on whether such a rope can even be built and deployed. It's a lot of material, and fragile to non-gravitational hazards. And you have to play it out towards its far end.
Human technology sure can't do this today. Maybe the fast track is to create the legendary paperclip-maximizing nanobots[2] and have them build and maintain a cosmic-length cable out of linked paperclips.
It's easy enough to imagine a hard sci-fi novella about doing this experiment, and also easy to imagine a range of actual student theses setting bounds on different aspects of the idea (even better to consider a rope in a hard binary, vs one in softer and softer binaries, with the BBH ultimately reaching cosmological radial separations).
[2] I'm drawn to a pattern of describing a gravitational-wave-shedding binary black hole as "barbell shaped", with a (notional) thin, zero-mass hand grip connecting the weights at the end. My intuition is that if we strengthen that connecting hand grip, we will generate a lot of gravitational radiation at the ends, justifying the idea that we rip one or both massive black holes out of position within their host galaxies, with the black holes ultimately colliding rather than separating with cosmic expansion. I see a host of problems with this intuition, though, that approximations up to numerical relativity could explore.
