I have a lot to think about, thanks. I don't want to let the Arduino produce the pulse, because it also has to deal with several interrupts. With every interrupt there will be some calculations, and I'm afraid this will mess up the pulse, making it useless for the receiver. And since the moment of transmission is relevant, I do not want to have it sent multiple times.
When using bit-banging PWM, it will be affected by interrupts and create jitter, but I don't believe this will be the case when using hardware PWM.
There is some code provided here. It looks like if we use 1 / (1/16MHz * 286) = 55.944KHz , this would be quite close to the 56KHz needed. How to create a 38 Khz pulse with arduino using timer or PWM?
You may also find this link interesting: Serial comm using IR
I don't think the 555 is needed - note that the 555 would normally generate a 50% duty cycle and this would only allow you to pulse at 2x the continuous current rating of the IR LED, which would be far lower than its peak current rating.