Thunderbird Lt Ecu

That sounds bizarre on many levels to be honest. Are you sure the '3rd party' is not Triumph HQ? I can't imagine anyone else has better ability to diagnose the ECU than triumph. Have the dealer tried flashing the ECU yet? By flashing I don't mean exposing themselves, but wiping and reloading the factory settings

I've never had an issue like that on my TB Commander. The longer I go without riding it, the more likely it is not to start - I've had fuel 'go off' in the tank, flat batteries, etc. But never not starting after it's been running, if it's warm it'll usually start on the button.

What happens when it doesn't start? Does it turn over, and just not fire up? Could it be your plugs or HT leads (not sure why that would mean a cold start works though!)?

If you were to stall the bike when riding, could you start it straight back up? or is it only when you turn it off by the key, and then back on again? If you don't know the answer, I suggest starting the bike, run it for a while, put it in gear, and slap the kickstand down, that'll stop the engine but the ignition will still be on. After that, take it out of gear and try and start without touching the ignition. If that works, but the same thing doesn't if you stop it with the ignition, it could be something to do with your injector priming. When you first turn the ignition (I think even if the bike had just been running), you should hear mid-high pitch whine, which is the injectors priming. To be honest, I have no idea how the injector priming could prevent a start, but ... it might give you a route to explore.