A couple of things up front: first, you seem to be talking about the present-era expansion of the universe, not inflation. Inflation, if it happened at all, ended about 14 billion years ago, long before the formation of large structures like galaxies and very long ropes.
Second, a galaxy with a recession velocity of c in the present era is going to be some tens of billions of light years away, not 1 light year. For the rest of this answer I won't mention any specific distances.
The short answer is that your string is a physical object and you haven't specified its properties well enough to determine what will happen.
One possibility is that the string is extremely stretchy (very small spring constant), and of low mass, and each part of it is initially comoving with the local Hubble flow (the averaged speed of nearby galaxies). In that case, the string will expand along with everything else.
Another possibility is that the string is essentially unstretchable (high spring constant). If it still starts out with each part moving with the local Hubble flow, it will almost immediately snap into many pieces. If it starts instead in some large-scale approximation of inertial motion, so that in the short term it doesn't snap, then it will continue to move that way for a while. In that case only a small part of the string can be locally at rest relative to the Hubble flow; the rest of it will be moving (relatively) toward that part of the string, with at least one of the ends moving (relatively) quite rapidly. But that doesn't mean it will break. In the short term, it's just like a very long thin galaxy with its own peculiar velocity.
In the longer term, the string could stretch and break under the influence of the cosmological constant, or it could collapse on itself, depending on its construction, its initial motion, and the local density of galaxies.
Regardless, the important thing to understand is that there isn't an expansion force making things move with the Hubble flow. The expansion of the universe is just the motion of the matter in it, subject to inertia and gravity. The reason that the galaxies are moving away from each other is that they condensed out of primordial matter that was moving apart. But your string didn't simply condense out of primordial matter; you constructed it somehow and moved it into place for the purpose of your thought-experiment. Its subsequent behavior depends on the details of how you did that, and not on what any of the nearby naturally occurring galaxies may be doing.
