Telescopes that use adaptive optics now use flexible mirrors to adjust for distortions in the atmosphere. The mirrors are flexed in real time to account for the distortions. I am wondering if it would be feasible to instead use fixed mirrors and use software to make the needed corrections to the image?
Short answer: no.
Why not? because the mirror is placed at a position in the optical train such that it controls the wavefront phase and tilt of each subsection, "subaperture" of the incoming beam. Once an optical system has formed an image, all phase information is lost. There are some software methods which can correct for simple aberrations such as astigmatism or coma, but those require a near-constant type of error across the input field. The problem with atmospheric turbulence is that the errors are essentially of a localized and random nature.
Note that even a plenoptic camera which, by tricky use of sublenses at the focal plane, record enough information to change focus, cannot correct for general phase aberrations.