I think it's possible in adobe bridge by making a contactsheet that is big enough
for all images in 1 folder and choosing your own layout
it's often called an "action sequence"
1) Export your layers to files and put them in a new folder
If you open that folder , you will see
2) Then you run a batch on that folder with the action you made, to remove the parts on each flile
3) ) Load files into stack , your new PSD will show all layers in the "Layer"- window
do you want a "batch" to do the same action on each layer ?
then you can File > Scripts > Export Layers to Files
and put all files in 1 folder
then you can run the batch and take this folder as source
after this batch, you can File > Scripts > Load Files into stack
1) use direct selection tool to select both parts
2) take pen-tool and move close to first point and see the cursor change , and then click
3) go with the pen tool and move close to the second point and see the cursor change , and then click
do the same for the other points