OK. Serviced the Tex (50k miles), put everything back together and went to fire up the throaty triple.

Lights flashed and needles swung as they should, immobiliser light came on, went off, then came on permanently!

Result? Not a peep, squeak or groan from the starter circuit, and the Tex just stood there, all forlorn.

I do not have an alarm fitted, just the OEM immobiliser.
Removed tank (again. Hate taking that off),

and checked carefully everything was connected and tight, especially the Immobiliser ECU. Reassembled, and nada! Same thing happened.

Took it to a good dealer, who believed it was either the keys becoming unmapped, or the immobiliser ECU failed. He tried everything - mapping existing keys, mapping new keys, checking comms between the 2 ECU's, and the diagnostic check just kept stating 'no keys found'.

Didn't really want to leave it there for a few days, as they were really struggling to work out what it could be.
Saw a complete lock set with keys, and both ECU's from a breaking bike (2014 Tex), and purchased the same, hopefully this will cure the problem!

Silly question - how to remove/replace the seat lock above the rear number plate? Can't see an obvious way to do this...
