### Godot version

v4.0.alpha11.official [afdae67cc] (when the feature was added) and newer (still happens in RC6)

### System information

Windows 10

### Issue description

I'm unable to handle when the user sets an exported node property (this feature: https://github.com/godotengine/godot/pull/62185) from a setter in the editor (tool script).

I don't know if this is intentional or not, but for me it was unexpected. I had to revert back to `NodePath`s.

### Steps to reproduce

- Create a scene
- Attach a script to a node
- Add `@tool` at the top of the script
- Export a node property with a setter, preferably with an easy way to see if it executed:
	```
	@export var node:Node:
		set (mod_value):
			prints("node", mod_value)
			node = mod_value
	```
- Save, close and open again the scene to make sure the tool script is running in the editor
- Set the node property from the inspector
- Observe that the setter didn't run

The setter is not bypassed when running the game, it is only bypassed in the editor.

We can compare with a `NodePath` property, which setter is not bypassed regardless if it is in the editor or in game runtime.

### Minimal reproduction project

[nodepath.zip](https://github.com/godotengine/godot/files/10840411/nodepath.zip)

switch (p_what) {
		case NOTIFICATION_THEME_CHANGED:
		case NOTIFICATION_ENTER_TREE: {
			Ref<Texture2D> df = get_theme_icon(SNAME("DistractionFree"), SNAME("EditorIcons"));
			open_big_text->set_icon(df);
			Ref<Font> font;
			int font_size;
			if (expression) {
				font = get_theme_font(SNAME("expression"), SNAME("EditorFonts"));
				font_size = get_theme_font_size(SNAME("expression_size"), SNAME("EditorFonts"));
				text->add_theme_font_override("font", font);
				text->add_theme_font_size_override("font_size", font_size);
				if (big_text) {
					big_text->add_theme_font_override("font", font);
					big_text->add_theme_font_size_override("font_size", font_size);
				}
			} else {
				font = get_theme_font(SNAME("font"), SNAME("TextEdit"));
				font_size = get_theme_font_size(SNAME("font_size"), SNAME("TextEdit"));
			}
			text->set_custom_minimum_size(Vector2(0, font->get_height(font_size) * 6));
		} break;
	}
}
void EditorPropertyMultilineText::_bind_methods() {