### 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)

option_button->set_disabled(p_read_only);
	edit_button->set_disabled(p_read_only);
}
void EditorPropertyTextEnum::_emit_changed_value(String p_string) {
	if (string_name) {
		emit_changed(get_edited_property(), StringName(p_string));
	} else {
		emit_changed(get_edited_property(), p_string);
	}
}
void EditorPropertyTextEnum::_option_selected(int p_which) {
	_emit_changed_value(option_button->get_item_text(p_which));
}
void EditorPropertyTextEnum::_edit_custom_value() {
	default_layout->hide();
	edit_custom_layout->show();
	custom_value_edit->grab_focus();
}
void EditorPropertyTextEnum::_custom_value_submitted(String p_value) {
	edit_custom_layout->hide();
	default_layout->show();
	_emit_changed_value(p_value.strip_edges());
}
void EditorPropertyTextEnum::_custom_value_accepted() {
	String new_value = custom_value_edit->get_text().strip_edges();
	_custom_value_submitted(new_value);
}
void EditorPropertyTextEnum::_custom_value_canceled() {