Hi Manta
I know this is a difficult problem to simulate or to find, it took me a few months.
A new battery is always a good place to start.
Starter Relay:
As I know it the starter relay is suppose to divert all the power from the headlights to the starter thus totally off not just dim. Mine flickered on and off before I replaced the relay, now it stays off the entire time the starter button is pressed.
If all the power is not diverted it could cause the motor to turn over slow as well.
I found a replacement at a good electronics store, just ensure the internal diagram printed on the side of the relay and the rating is exactly the same as the one you removed.
Charging System:
Did you follow the whole procedure to test your charging system as specified in this link:
http://www.triumphrat.net/speed-tri...-diagnostics-rectifier-regulator-upgrade.html
Therein it specifies testing the stator and rectifier as well, 12.9V is not bad, Mine chargers at 14.5V when cold and when after a good ride the voltage decreases to 13.45V mainly due to the battery being fully charged.
Starter Cables:
It is a good indication that all 3 cables measure 0 Ohms when cold, it might be that the problem is aggravated due heat soaking. Feel if the cables are warm after a ride and if possible try testing the resistance values after the ride to confirm if the resistance value have increased...
Connections:
Have a good look at all your connections especially the engine earth terminal, battery terminals, solenoid connections and starter base and bolts. These form the starter circuit and rust and any other material on the terminals or connection points could hinder a good electrical connection. Ensure that these terminals are clean and tight.
Also check your wire connections into the fuse box and fuses.
Solenoid and starter:
These can be both bench tested, I bench tested the solenoid and serviced the starter.
My starter had a lot of carbon on the communitator from the brushes but easy to clean off with some electrical cleaner and fine sanding paper.
Please note the starter can also be affected due to heat soaking as the resistance of the rotor windings will be affected due to heat.
I truly hope you solve your hot starting issue, but when it happens be patient and wait that 5 to 10 min before starting the bike again.