With this method, you get the force that the flag exerts on you while riding the given speed. Now simply multiply with your speed, and you get the power that's drained by the flag.

The advantage of this method is that you can see when the run is spoiled and should be rejected. That is, you can spot a change in wind, or a passing car, or a change in other conditions. The sample size is the length of the run in seconds, not the number of total runs. You do want to have a wide variation in "entering" speeds so the runs are statistically independent and you can "pry apart" the estimates of CdA and Crr.

This method is a special case of the Virtual Elevation method where one uses an accurate power meter to collect the data. In this special case, we know that power while coasting is zero, so accuracy in power is not an issue. However, as in all estimates using Virtual Elevation, you will want your speed measurements to be as accurate as possible, so it's best to use a separate wheel sensor and to measure wheel roll-out accurately. Measurements of CdA attained using Virtual Elevation have been validated with wind tunnel measurements, and VE is the basis for most of the field tests done by pro cycling and Olympic teams. They use power meters not because the method requires it (the mathematics is exactly the same) but rather because data collection is faster: you can go both uphill and down, and it's easier to get a broad range of speeds since you can pedal. In typical coast downs, you are limited in the range of speeds you can easily attain and the length of uphill segments is short.

If we want to take into account the drag of the pole, then I suggest attaching the bottom of the pole to a pivot with a guide so that it tries to tip backwards, then using a strain gauge to prevent it doing so, mounted above the pivot. In this case you could actually use a spring balance instead of a strain gauge, remembering that they measure mass not weight (in physics terminology). At this point the leverage comes into play as well. Wikipedia has the leverage equations and some background, and you need to treat the pole and flag as separate sources of torque. You can probably just about treat the drag on the pole as a force at the centre of the bit exposed to the wind, and the drag of the flag as acting at the top. You could measure the force on the pole without the flag, and then the force from both, subtracting the first to work out the torque due to the flag.

The force is proportional to the square of the wind speed (or simply the speed as I've suggested a still day) so it would be better to multiply the instantaneous values and not just work with averages.

Estimating aerodynamic drag can be tricky, but can be done with field tests. If you can find a quiet road that is protected from the wind and you have a way to record speed each second, you can do coast down runs to determine the drag with and without the flag. The approach is discussed on page 108 here. You do need a recording device to capture speed each second, but nowadays many riders use bike computers with this feature. If you have a device such as this and a speed sensor on your bike, you need no other special equipment.

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I think the trick will be to measure (ideally log) the force and multiply by the ground speed on a still day. An answer to an old question of mine has some useful formulae for reference.

I have a recumbent bike that has a safety flag on the top of a thin pole, so it flutters around higher than and after my head.

Note: Work with the correct units. 1kg weights 9.81N, and if you multiply the force in N with your speed in m/s, you get Nm/s = J/s = W. Just make sure that you convert the inputs to the multiplication correctly.

With the bike still, pull the flag back, measuring the force somehow. You can use a spring scale, or just hang a bottle with different amounts of water on a string, connect that string to the pole, and run the string over pulley or a round, smooth piece of metal.

If we assume only the flag contributes to the drag, and not the pole, then connecting the flag to the pole with a strain gauge would work.

Once you have CdA and Crr, you can estimate the power "cost" in terms of watts from your flag at different wind speeds, not just at a given speed. The conversion formula for converting to power is given in one of the answers to that previous bike.SE question.

If you can't log a strain gauge electronically, or you use a spring balance, you can always video the display with a clock synced to your speedometer.

Details are provided in the link but, in summary, you will want to do at least two runs starting at different speeds, then solve for the two unknowns of CdA (drag area) and Crr (coefficient of rolling resistance) given that the road surface and road profile are the same for both runs. Since your Crr should remain constant across runs, that will give you another constraint to use to evaluate the estimated CdAs.