Oh my, how would I even do that
A cheap and precise way to measure active power + many other useful parameters is to re-purpose a surplus electricity energy meter (real commercial one, not plug-in toys) . Usually, a DSP chip inside calculates power as an integral of momentary U*I over a few line periods, at sample rate of about 2kHz. This is pretty enough. 24-bit ADCs are typically used, and the precision mainly depends on calibration quality of voltage divider and current shunts or transformers.
Caveats:
On most meters, measuring circuit power supply is feed from voltage input. This power is not high, but will somewhat throw-off measurements on smaller lamps. May be easily hacked to an aux input, and some more elaborated meters have jumpers for selection from where the power to the supply goes.
Ignition pulses need to be taken into account. A good approach is a SPDT switch which disconnects and shorts meter input from the circuit until the lamp stabilizes.
Input voltage range must be respected. Sometimes, MOVs are placed across the inputs and will burn if overdriven. Also, ADC inputs may start to distort. Feeding lower voltage is usually OK, as ADCs used are very linear across wide range, though the software may start thinking it is a brownout and stop measuring.
