This what the selection tools are for. For instance you use the lasso tool to draw a polygon around the face, add some feathering (Select>Feather or use the tools' feathering option), and then do the color changes (hat will only apply to the selection area).
Layer > New Layer or Shift+Ctrl+N (make sure it's transparent obviously)
Select > All or Ctrl+A selects the entire layer. Or if you want to load a shape drawn on a transparent layer as a selection, right click the layer in the layers panel and choose Alpha to Selection. Obviously you will then have to select the transparent layer again before ...
You can try a Displace map. Grey pixels in the map don't move, dark pixels move to one side, light pixels move to the other. By using a black-to-white gradient where the grey is on the middle of the can you can stretch the extremities more than the center:
You can use Curves to give your gradient more effect on the sides:
But getting the original flat ...
A good way to demonstrate the problem is to blur a ring without a selection. With Gimp's Gaussian blur the result is slightly diamond-shaped:
There is a very powerful collection of filters available for Gimp called GMIC. It features (among many other things) a Gaussian blur filter that doesn't seem to suffer from the problem (at least not to the same extent)...
The answer turned out to be blindingly simple: Just select the layer and then apply the filter. I did try that before asking, but for some unknown reason it didn't work at first. In the end, however, it did.
If your image is in grayscale or color-indexed mode (see the title bar),
anything you add to it (cut/paste, or Open as layers) is coerced into its current mode (for grayscale: desaturated, for color-indexed: converted to the closest colors in the colormap).
You can restore full color support in your image by putting in it "RGB" mode: Image>...
Use a different method instead. Use the Clone Tool.
Select the Clone Tool, choose a small soft edged brush
Ctrl+click to select the clone source from a different part of the bag
Carefully paint over a small part of the finger
repeat steps 2 and 3 until you have finished
Such transforms are mathematically difficult to define. If your transform is a sine/cosine law, you can use the filters that project your layer to a cylinder (Filters>Map>Map object and map to cylinder).
For an arbitrary transform, possibly using a displace map, but the actual map may be difficult to come up with:
To create the displace map (#2):
There's in the basic GIMP configuration filter Distort > Spherize. In horizontal mode it deforms your image like it was used as a bottle label. It's definitely non-linear but you have only a little control. A custom math transformation would be better. It's discussed at the end of this answer.
Unfortunately the sparsest area is in the middle, not in the ...
I assume that after "Channel to selection" you bucket-filled the selection with I white on a black layer.
On Gimp 2.8 the behavior is as you expect(*). 2.8 works all over with 8-bit gamma-corrected values (so that there are more values to describe the dark tones).
In 2.10 things are different. Internally all computation is done in floating-point, ...
There's no need to make a selection first. Change the Threshold slider in the tool options until you get the desired result. Also you appeat to have an old version of GIMP. Perhaps consider upgrading to a more recent version if you want this functionality.
Example in GIMP 2.10.18
Selection by color isn't sharp because at the edge the color often changes gradually. You must bite the bullet and draw a path around your shape. It makes as sharp selection as your photo allows. There's plenty of tutorials available like this: https://www.youtube.com/watch?v=SqUineYZmpA
BTW I have made numerous selections in photos to extract items and ...