Hello and welcome to the site from New Jersey, USA, Captain Pelling.
Sorry to hear of your inauspicious start to what will hopefully be a long-term love affair with the R3R.
My two cents is that you purchased a brand new, factory fresh motorcycle that is grievously defective (musta' been a bad day on the production line at the Hinckley plant). As you noted, repairing it entails the return of a more than slightly used and abused machine to you. I believe that you have a strong leg to stand on with regard to demanding of Triumph a new and properly functioning R3 to replace the defective unit that you purchased in good faith. Depending on the consumer protection laws in place where you live and since your new motorcycle came with a factory warranty protecting you against manufacturing defects, that is the tack I would take were it me in your place. If your dealer won't fight for a replacement, then you will have to do battle with Triumph directly, using the law to your best advantage and leveraging Triumph's own warranty against them. In instances such as yours, they seem to do the right thing more than they don't, especially if yours is a factory-related flaw.
The rest of my advice, for what it's worth, is best derived by clicking on the YouTube link at the bottom of this post and listening very, very carefully to the lyric. Godspeed!