As already stated, you can't save blending modes to a .png. You can try to imitate it, but there are some severe limitations.
As Circle B suggests, you could use a regular gradient, the limitation of this is that it will inevitably dull any colour behind it, hence making the background look dirty rather than illuminated:
(Comparison shows your dodge on left, then the transparent png on the right)
You can get a little more creative with the transparent png, but at the expense of severely limiting the utility of the image:
If you look at the png on a black background you can see there are a bunch of major limitations:
- It is specific to one background.
- It can't be resized.
- It'll only be reusable on patterned backgrounds since it must be aligned with the repeating pattern of the image (in this case at intervals of 13px horizontally, and 12px vertically)
If you did think that technique could still be of use, you'd do it like so:
- Create the background layer.
- I took your dodge layer from the question (set to "Dodge")
- Duplicate the background, place it on top of the dodge layer and set it to "Subtract".
- Merge those layers so they become rasterised
- Use Color -> Color To Alpha, and select black as the colour.
But I'm not convinced there would be a good reason to..