Technically, what you're describing is called tethered formation flying. Dynamics of such systems are rather complex, but it's simple to say the way you proposed it in your example wouldn't work without constant propulsion to account for different orbital decay rate and attitude control to maintain their rotation rate along the velocity vector constantly aligned so it's oriented towards each other along the tether.
This would have to be done on both ends and constantly, so you don't gain anything from maintaining such a system unless you really need the two satellites physically connected. Say to remove the need for them to communicate wirelessly with each other, which could be tapped into or the non-tethered constellation detected via compromising electromagnetic emanations of its members. Or the lower altitude end serves as an anchor mass for electrodynamic tether, and generates sufficient current that their attitude can be maintained via, say, magnetorquers.
Most striking obstacle from your own example however is that the atmospheric drag on the lower altitude satellite will be significantly larger than that on the higher altitude satellite. For the whole tethered system that means its orbit would decay significantly faster, with the lower altitude satellite dragging both of them into even lower orbit, removing the whole system's specific orbital energy.
In simpler terms, pull of the lower altitude satellite on the higher altitude one via tether and along the velocity vector axis, or v-bar, and making the lower one lag behind if they weren't tethered, would decay the system's orbit faster. Constant tug on the tether would inevitably pull the higher altitude end lower than what used to be our lower end of the system by simply increasing its orbital eccentricity to the point that its periapsis is below the Earth's surface half an orbit around the Earth later. In a sense, it would run aground, assuming it didn't burn up in the lower atmosphere first.
Smaller scale tethered formation flying is possible though, and if they're close enough to one another, the distance between such formation's members can also be maintained by electromagnetic force alone. See e.g. Department of Defense (DoD) SPHERES-Resonant Inductive Near-field Generation System, a.k.a. SPHERES-RINGS experiment, also demonstrated during one Space Station Live (YouTube video). The experiment is meant to test wireless power transfer and formation flight using electromagnetic fields. Before the link disappoints you though, we're talking of stretch of an arm distances here, not nearly two thousand kilometers between them.
Also see other types of momentum exchange tethers. Perhaps the most similar to what you're after could be considered the skyhook concept, and what I describe would happen to it if it was following orbital altitudes from your example, for a moment at least, would be similar to the rotovator concept, with the exception that its final periapsis would be too low and you'd end up with 1,840 kilometers of extremely strong and thin cable crashing at its terminal velocity towards the densely populated equatorial regions.