Yes, turning off compatibility on layered PSD files will reduce the size. But the Perspective Cropped version will still be considerably larger for some unknown reason.
To get smaller files, either:
1. Don't use PSD, use Tif instead. No problem there.
2. If you continue to use PSD, don't unlock the Backgdound layer and make "Layer 0". No need to do that. Of if you do, flatten the file before saving.
I've never used Perspective Crop before this. I use Edit/Transform instead. But as I noted above, even Edit/Transform causes the strange file size increase under the right conditions. No big surprise since they both do the same thing.
I studied the PSD vs. Tif debate and did some testing long ago. As a result, I use Tif with no compression on 16 bit files and Tif with LZW compression on 8 bit files. That, for me, is the best compromise between speed and size. The Tif with ZIP compression makes just slightly smaller files, and works OK on 16bit, but the ZIP compression is much slower than LZW.
One of my test examples:
A file saved as 16bit layered PSD with no compatibility was 109MB and saved in 9.8 seconds. The same file saved as 16bit Tif no compression was 120MB and saved in 0.2 seconds. I choose to trade the Tif's 10% file size increase for the 490% speed advantage.
The same file saved as 8bit was a 62MB PSD and a 36MB Tif, and the Tif saved much faster of course. No contest on 8bit files.