There’s just enough slack in the LED wiring to clip a Tek current probe in there:
Which reveals the LED current waveform:
The LED is on continuously, except for the two 75 ms Morse code dits in the upper trace.
The lower trace shows the current ramping up at the end of the first dit, from zero to 400 mA in 1.5 ms.
Clamping the probe around the 6.3 V power supply lead:
The supply current includes maybe 20 mA for the Arduino running the Morse code program and the current ramps up from there to about 250 mA when the LED is on.
The LED drops 2.6 V at 400 mA, so it dissipates a smidge over 1 W. The 2.0 Ω current sense resistor (3.3 Ω in parallel with 5.1 Ω) dissipates 800 mV × 400 mA = 320 mW.
The dissipation from the Bafang headlight output, including the Arduino, is 1.6 W.
The running light ticks along at the hot side of comfortably warm on the Electronics Workbench and runs barely warm in free air out on the bike, so I’ll define it to be Good Enough™.