What should I look for when diagnosing electrical circuit gremlins where load is much less than capacity?
Replace Capacitors in LCD TV Power Supply

I would suspect that the problem is within the LCD TV. The power supplies inside LCD TVs and LCD monitors use cheap electrolytic capacitors that go bad and cause problems with the device. Your symptoms are representative of the TV power supply just starting to go bad. Small amounts of transient noise on the AC power line caused by switching other loads on and off, particularly the fan speed controller, gets into the TV power supply and causes a disturbance such that the various DC voltages inside are momentarily going unstable. When the TV was new the capacitors in the TV power supply were able to filter out the noise and power line disturbance but as they start to go bad the TV acts up. 

It is highly likely that you will continue to see degradation in the performance of the TV over time. A common set of symptoms to look for include:


The TV may become difficult to turn on wherein you have to press its ON button multiple times to get it to stay on.
In the failure progression you may observe a point where when you go to turn on the TV it will blip on for 1 - 2 seconds and then go back off.
Eventually it may get to the point that the TV will just simply fail to turn on at all. 


I have personally experienced this problem with two of my own LCD monitors and got them back to perfect working order by replacing the power supply capacitors. In this case the monitors were about 3 years old. At my place of work I have been reclaiming failed monitors, ranging in age from 2 to 6 years, that were set for scrapping simply by replacing their power supply capacitors. So far I've reclaimed over a dozen monitors that are now keeping their owners happy as 2nd or 3rd monitors off their company issued laptops. Most recently a colleague came to me with his LCD TV with the problem that it was failing to turn on. I replaced all the electrolytic capacitors on his LCD TV power supply board and he reported over the Christmas break that his TV is now working perfectly.