How to stabilize exposure in After Effects?
The under exposed section will never have as much detail in the shadows as the correctly exposed section, so you will never be able to make the second section look as good as the first. Having said that:

You need to duplicate your layer, then correct the exposure in the second half using level controls, or colour correction effects. 

As you correct, switch the layer on and off, to compare it with a good part of the first layer. 

Once the second half is looking better, then work on the transition point. 

The second layer should be on top of the first layer.  

Make some Opacity key frames on your second layer, starting at 0% and building up to 100% when the poor exposure section has fully begun. 

You may find that a dedicated application like Davinci Resolve (the free version) is better for colour correction than After Effects.