Illustrator has no equivalent to Photoshop History Panel features, including the "Snapshot" feature.
The best way to utilize the same assets multiple times without direct duplication in Illustrator is via Symbols. But, Symbols are quite different than any "Layer Comp" or "Snapshot"
You cannot put gradient meshes or raster images on a perspective grid. So first you have to rasterize the gradient mesh (Object > Rasterize), and then you have use an envelop distort to get the rasterized image in perspective. Here's how you do that:
There is.. https://youtu.be/0pRemmx-kKQ?t=26
Open your original image and the image where you want to based your original's pixel colors.
Then in your original, go to "Image > Adjustments > Match Color..."
Photoshop has a Match Color adjustment available under Image > Adjustments > Match Color, however the results may not be quite what you expect.
I'll let you be the judge.
You might get better results by sampling colours manually from the source image, and manually applying them to the drawing using the Fill Bucket tool.