Yes, you can use TuneECU on your TBird Nolton - any Triumph actually - just plug in your cable & read (& reset) the code.
I think it's always important to at least see what code it is and then determine what you need to do about it.
Incidentally you could disconnect battery, put ECU on shelf for a year and plug it in and it would STILL show code.

Even if light goes off after three full warm-up/cool-down cycles (not just starts on/off) - which would mean whatever fault tripped light initially is not a current problem - the code that caused the trip would still be in memory (and readable by TuneECU) for 40 more warm-up/cool-down cycles.