Which method should be selected to enhance the compressed image?
I would advise you to try between two different approaches, then test and see which one performs best.

The first one is intensity clustering. In this approach, you data points are one-dimensional and correspond to the intensity of your raw pixels. Then, run a clustering algorithm such as $k$-means with $k = 256$, and perform the final quantization by replacing the intensity of each pixel by the mean value (or the index) of the corresponding cluster mean. With this approach you can limit the information loss by forming an image of indices and keeping a look-up table of the real (14 bits) cluster mean values.

The second would be to perform some local histogram equalization, i.e., cutting your image into blocks and performing equalization (or any other enhancement procedure) on each block.
You will obtain an image where the features are mstly preserved and that you can remap linearly into $[0,255]$.