Then, I used a led resistor calculator (the parallel calculator, since I suppose the LEDs of my display are mounted parallel)... which tell me that I need a resistor of 1 Ohm (AKA: I don't even need a resistor), when the input voltage is the same as the forward voltage of the LED, and doesn't matter how much current I enter as the "desired LED current".
Because these calculators are very simple minded and do not take into account extremes like you have in that situation.
As well as considering the average current then you have to consider the peak current. This is the maximum instantaneous current that will flow in any given situation. With PWM this peak voltage is 5V no matter what the duty cycle gives you. Therefore you have what 5V will drive through the LED.
This will be limited by the impedance of your source, in this case the arduino pin. However remember that the data sheet says the absolute limit you should draw from a pin is 40mA. Anything at this level or over this level will damage your pin. You might not see that damage immediately or even in a month or two but the circuit around that pin will die sooner than it otherwise would.
In my tests with an arduino I have found that you can get 250mA peak current from an arduino pin, more than enough to damage it and possibly damage any LED connected to it.
Now you can totally ignore the data sheets and do what you want and your individual device may or may not fail immediately, maybe next month, next year or never. But if you take a large number of devices abused in this way some will fail quite soon. Damage is cumulative and the impedance of the output will gradually increase, maybe too slow for you to notice it without making measurements but it will occur.
On the other hand you might think we are all in a conspiracy to make you spend an extra $0.01 on a resistor, funded by the resistor manufacturers. If so I want my cut now from all the unnecessary resistors I have encouraged beginners to use. On the other hand have you noticed that only beginners as this sort of question?