Just taking a guess here, you might want to try replacing the standard flasher relay with an electronic one. The standard relays are very sensitive to current differences. The electronic units should flash the same no matter what wattage is drawn.
Here's an article I found about car flashers, but the principle is basically the same:
Fix your Blinkers
Triumph turn signal flashers are heat activated switches. Activating the turn signal switch allows an electric current to flow through a strip of metal in the flasher module which acts like a heating element. As the metal strip heats up it bends and disconnects the circuit. With the circuit disconnected it begins to cool down. Once it cools down it returns to its original shape and restores the current flow, repeating the cycle. This series of hot/cold cycles makes your signal lamps go on and off.
The higher the voltage across the heating element the faster the metal strip will heat up. A dying lamp or a bad connection somewhere in the circuit will cause the voltage to drop and the flashers will start to cycle very slowly or remain on all the time. The left and right circuits share a single flasher module. If your right turn signal works fine but your left turn signal blinks very slowly then you probably have a bad bulb or a bad connection somewhere in the wiring for the left side. If both circuits fail then problem is either in the portion of the circuit that is common to both sides or the flasher module itself is going bad. Sometimes a bulb will create a short circuit when it fails. A short circuit can cause a much faster blink rate and will not always blow the fuse.
My flasher modules were failing about every six months. The new modules all seemed to come from Asia and they were not of the same quality as the original Lucas flashers. I don't know if they're more susceptible to vibration than the old ones or if the metal strips are weaker and are failing after a certain number of cycles.
You should spend the time to replace old lamps and clean the electrical contacts but flasher modules will still fail eventually. The permanent solution to the problem is an electronic flasher module. Electronic flasher modules cost between $10 and $15 at most auto parts stores. They use a solid state timer to regulate the flash cycle and are not affected by low voltage. Flasher modules come with either 2 or 3 terminals. My TR6 uses a 2 terminal module located in the passenger foot well. A second 2 terminal module near the fuse box is used for the hazard flashers and can also be replaced with an electronic module. Other Triumphs uses 3 terminal modules so check yours before you head to Auto Zone. The only downside to using an electronic flasher is your blink rate will no longer change when there's a problem in the circuit. If you have an electronic module installed you should periodically check all the signal lamps to make sure they are working, but you do that anyway, right?
