It's correct that the inrush current is the MAIN problem with LED lighting.. Though it should be noted that most dimmers now actually circumvents this problem by trickle charging the LED-driver capacitors, and by doing this, can absolutely handle the continuous wattage specification.

It's quite easy to check if you have such a dimmer (or LED-driver for that matter).

Turn the lights off. Let them stay off a couple of seconds.

Turn them on. If they don't respond IMMEDEATELY when the button on the dimmer is clicked (like they do when shut off), the dimmer is "trickle charging" the capacitors with it's own circuitry. When the inrush current goes below a threshold that is acceptable, the lights will turn on and "full power" is fed to the lights.

This can also be noticed if a LED "gradually gets bright", and not "direct full brightness" when it's switched on.

If it's a very old control device / dimmer, it might not have this feature, and on those, you'd be best of by dividing it's rating by at least 10.

Is the fan maybe humidity controlled? Any small dials / knobs that are visible on it's back? We had such a bathroom fan, and the humidity detector was faulty, but luckily, twisting the "sensitivity dial" to max, just kept the fan on continuously.

Other than that, does it spin if you just push it? As in; is it able to rotate freely?

Might also be nice to know what make and model it is if that's available =)

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One thing that stood out, was that you measured only .3 volts at the switch? as in 300mV?.. That's absolutely nothing, so either there is no available power there, or something on that line is short-circuited (which would drop the potential to nearly 0 volt) but that would also cause any fuses/circuit breakers to pop.