### 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)

if (!big_text_dialog) {
		big_text = memnew(TextEdit);
		if (expression) {
			big_text->set_syntax_highlighter(text->get_syntax_highlighter());
			big_text->add_theme_font_override("font", get_theme_font(SNAME("expression"), SNAME("EditorFonts")));
			big_text->add_theme_font_size_override("font_size", get_theme_font_size(SNAME("expression_size"), SNAME("EditorFonts")));
		}
		big_text->connect("text_changed", callable_mp(this, &EditorPropertyMultilineText::_big_text_changed));
		big_text->set_line_wrapping_mode(TextEdit::LineWrappingMode::LINE_WRAPPING_BOUNDARY);
		big_text_dialog = memnew(AcceptDialog);
		big_text_dialog->add_child(big_text);
		big_text_dialog->set_title(TTR("Edit Text:"));
		add_child(big_text_dialog);
	}
	big_text_dialog->popup_centered_clamped(Size2(1000, 900) * EDSCALE, 0.8);
	big_text->set_text(text->get_text());
	big_text->grab_focus();
}
void EditorPropertyMultilineText::update_property() {
	String t = get_edited_property_value();
	if (text->get_text() != t) {
		text->set_text(t);
		if (big_text && big_text->is_visible_in_tree()) {
			big_text->set_text(t);
		}
	}
}
void EditorPropertyMultilineText::_notification(int p_what) {