After some testing I conclude that a cr2032 is not a good choice for powering this project. A really large rail capacitance would be required to get a high enough current through the LED over the duration of the transmission. The losses on capacitor leakage would be unacceptable.

Voltage measured over a 1Ω resistor in series with the LED. The first few pulses measure at around 500mA - just as planned.

Unfortunately, even 800μF worth of caps wasn't enough to prevent the rail sag.

The 300mV drop on the power rail corresponds to a 300mA lower current through the LED. The last few pulses are delivered at just 200mA. The range from which the message can be read clearly is around 10 meters, and requires pointing the receiver directly at the transmitter.

And that's with delays between pulse trains increased twofold, halved pulse train length, and carrier duty cycle of 25%.

I think that either 2 LR03's, or 3 NiMH 40H's with small solar panels will have to be used in this project.