I don't have a good answer for you, but I have tried something similar in the past and found it impossible.
The way Godot treats text, each letter has a very small, tight bounding box, and the glow effect gets cropped around each letter.
If you manage to get an outline shader working you'll have a good starting base, since outlines are similar to what you want, except for the soft gradient. Example of outline shader: https://godotshaders.com/shader/2d-outline-stroke/
You'll see that it doesn't really work with text. Note that Godot has an special "outline" parameter for text; I believe this is a developer's hack to get around this limitation.
Another approach would be somehow duplicating the text as an image, perhaps rendering it to a viewport texture, modulating and blurring said texture and showing it behind the actual text.