You can't just "peel off" a reflection - it's an integral part of the image you have. You need to dig in your heels and do it manually. Yes, it's doable, but it can take an incredible amount of time.
Which is why, in my job as an art museum photographer, a large part of my work consists of killing reflections before they hit the camera sensor. For that I use things like cross-polarized lighting setups as well as careful modeling of light sources and directions.
The best way to deal with it IMO is to set up a couple of adjustment layers, some lightening, some darkening - and then inverse the mask to black (to hide the adjustment), and gradually paint in the correction. Hit X to paint back out if you overdo.
Not trying to discourage you, just being realistic. This is difficult work.