Electronic – How to digitally “amplitude modulate” an LED

I don't think there is a precise answer, especially without seeing the datasheet for the device. Usually PWM is used for efficiency, dimming, or over driving. Over driving is what you're interested in, this is where you pulse a device with too much current, but before the heat destroys it, you turn it off so it can cool down. But like I said, you need to see a data sheet to know what it can take. See this question about High Current Pulse on LED.

I have tested several large LED arrays before and I think they hit the sweet spot with regard to lumen vs power (measured with test equipment, not perceived brightness,) at a duty cycle around 85-90% at a frequency greater than 600hz and less than 8khz, but I cannot recall what it was.

there must be a sweet spot where our eyes cannot identify there is a reduction in total power even though there is. For example if you Pwm the laser with 10khz with 25% duty cycle this is as good as dc to our eyes as we cannot see the on off cycle.

To clarify your thinking, as to what our eyes perceive as not flashing (Persistence of vision), anything over 16 flashes per second should be sufficient as long as it's not moving, hence films are 24 frames per second (at least they used to be.) However the eye perceiving it as solid light is a very different thing then not being able to notice brightness. When looking at a led pulsed with a frequency sufficiently rapid, the human eye can no longer resolve it as flashing, but it perceives the time average intensity; meaning you will notice a difference in the brightness. At some point the eye can't notice a change in brightness, and you will probably damage the eye, however that is if you are looking straight at it. The brighter the laser is, the farther and more noticeable it will be.

Make your preamp with a variable gain. Determine the desired gain experimentally. As you would imagine, the desired gain would depend on: distance, sensitivity of the detector, focusing lens on the transmitting LED.

You optical communication should work fairly well indoors and outdoors at night. Making it work in daylight may be a challenge. The Sun can saturate the detector.