@Elfwyn Doesn't look like that's possible, at least right now as these are the steps to redeem the code and choose your name:
Your name here in the community is separate to the your Stadia name so don't worry about that!
I guess I have not chance but to ask around then if a friend of mine can lend me their phone....
Additionally, if you want to configure the controllers via Bluestack you will also need to have a Bluetooth controller available (as the initial setup of the controllers is performed via Bluetooth until WiFi is configured on them).
You need to setup a suitable location within Bluestack because location services are used by the Stadia app as well.
Of course if you are setting up Chromecast too, then you will also need to install the Google Home app as well.