Injection

Unless you want to start messing with the FI system it sounds like a trip to your local dealer.
Triumph set these bikes up to run lean and my Street Twin always sounds like it's about to die while warming up after the first start of the day. When warm it's not so bad. I have heard of other bikes in this range cutting out at tickover, so you're not the first.
I'd be interested to know how you get on.