You can't do that with a PNG.
What you need there is to make some transparent pixels luminescent, and that's only possible with an associated (sometimes called "premultiplied") alpha channel.
PNG uses unassociated alpha. That means that the alpha over operation will multiply the alpha on the foreground plate as part of the operation, destroying the information that sits on the transparent pixels.
The only way to solve this is to use a format that allows associated alpha channel.
Keep in mind that programs like GIMP (and Photoshop used to, but I'm not sure if it still does) internally convert images to unassociated alpha, making compositing that kind of images imposible.
The same happens with graphics for the web if you're going to use PNGs.
Blender's compositor and other compositing programs will allow you to do that, but if you need to use it in one of the programs mentioned above, you'll have to export the glow plate as a separate file and add it over the composite.
Here is a simple example that demonstrates how easy this is to accomplish correctly: