First of all, when you read about a "50W LED", it is certainly saying that it outputs the equivalent light of a 50W incandescent light (although that is often a exaggeration). That "50W" LED actually draws less than 10 watts. This means that it draws much less current and emits much less heat, both of which are the factors that typically limit the allowable light in a fixture.
You'd be able to replace a 50/100/150W 3-way with a lower wattage, even if staying with incandescent.
Second, unless the LED light is marked "dimmable", don't try to use it on a dimmer. You may damage the dimmer, light, or both. Even if the light is dimmable, some dimmers will not work well with a low load: a dimmer that works on a 100W light bulb may not work on an equivalent-light-output 13W LED. They make LED/CFL specific dimmers that may work better.
As for the 130V R40, that is probably a "long life" bulb. Running an incandescent at lower than rated voltage extends its life. Therefore, they make 130V bulbs to run longer on 120V. Just use LEDs rated for 120V.
As for your existing florescent lighting, I assume you're talking about the straight tube type. For a fixture that old, it's probably less efficient that newer florescent fixtures or LEDs but still much more efficient than incandescent. This is harder to replace with LEDs, just due to the area covered. There are LED tubes made to replace florescent tubes but they are still expensive and generally require re-wiring the fixture. These are most often used in commercial applications.
The fan controller is monitoring the lamp circuit to know when the lamp turns off. It seems as though it relies on a low-resistance path to neutral through the lamp filament to run the timer.
The LED lamps have fancier electronics in them so there may not be a direct path to neutral if the electronics is not running.
Best Answer
If all 3 incandescents worked previously without issue (so you thought), then I'd have to say that the dimmer is limiting the current. LED's are very sensitive & require very tight limits to operate. Meaning, the first 2 bulbs are reducing the voltage & amperage to leave the third bulb unable to activate.
Your 3rd incandescent may have been dimmer than the other 2 or all 3 may have stepped down equally. Which leads to, put an incandescent in the 3rd sockets & see if it lights, it should. Then, try incandescent & LED combo's to see what does & doesn't happen.
You may be able to adjust the dimmer, very few have this ability & you'll need the dimmer's manual to do it. But, I'd replace the dimmer with a regular switch to confirm that the dimmer is the problem. If so, then get a new dimmer or keep getting them until one works. I say that because I doubt there's anything on the box to indicate any difference from 1 dimmer to another.