A possible solution for this is to instead mount the mirrors on a fixed surface. There have already been proposals for building large telescopes on the far side of the moon, shielded from terrestrial radio interference. NASA has even demonstrated that large astronomy-grade mirrors can be constructed in-situ from lunar regolith. A telescope of this scale would have many applications and purposes besides viewing exoplanets, but this would be its primary purpose.
I would like to propose that instead of the lunar far-side, a better location for an optical (and maybe radio as well) telescope would be inside the basin of Peary crater. I've discussed the unique properties of Peary in this blog previously, in the context of human colonization. But briefly, Peary crater, by virtue of being situated on the north pole, has the triple virtue of a basin which is constantly in darkness and protected from solar radiation, a rim that is in constant sunlight for solar power, and a substantial amount of water ice on the floor of the basin. Shackleton crater on the lunar south pole is similar in many respects, and would similarly be an excellent site for this; however Peary is about 80 km in diameter and Shackleton is only about 20 km, so Peary would afford space for a much larger telescope.
Because the basin is always "aimed" at the moon's northern sky, and the same part of the sky is always visible throughout the month and throughout the year, very detailed long-term observations could be made unhindered of this part of the sky. Yes it would be limited to only this part of the sky; however, many space-based telescopes, including NASA's Kepler Space Telescope, have similarly limited fields of view.
And yes, a small astronomical observation outpost and research facility could provide the seed of a human colony in Peary crater as well. In later posts we'll be discussing the advantages of establishing a lunar colony over either Mars, Venus or orbital stations; a moon-based space telescope (at whatever location) could be a very good beginning to a robust lunar city.