Hi Gunther,
I have had a lot of experience using flash on birds. There are two basic situations (i) flash as main light and (ii) flash providing 'fill' light. With the latter, the flash is providing only a fraction of the light required for proper exposure. The total exposure is made up of a contribution from ambient light and the balance is provided by the flash. This situation is of no concern as the bird's usually have their eye accommodation adjusted for the ambient light and the extra contribution from the flash raises the total amount of light received by the bird up to about what would be expected if the bird were in good light for photography. Some birds are very sensitive to sharp noises and movement in their near environment and may 'jump' whether one uses flash or not. The noise of the shutter and the fast movement of the lens aperture blades even before the shot is captured can startle some birds. For example, I have Eastern Rosellas regularly visiting my garden and I often have recorded them jumping in response to the noise of the shutter and aperture adjustment (no flash). The point I am trying to make is that even though some birds are startled when using flash, it is not necessarily anything to do with the flash.
The second situation, with flash as main light, is more likely to be a problem. In this case, with nocturnal birds on a dark night, the eye will have the pupil dilated and the bird may experience some discomfort or temporary blindness. I say may because in my experience I have never actually observed owls and frogmouths show any adverse effects, in fact they normally just go about their business paying no attention to the flash firing. Better Beamers and similar light concentration optics are not recommended, especially at close range. I have used these devices but find them un-necessary with modern CMOS low-noise sensors and conventional flash units.
When shooting owls and frogmouths it is usually easy to see the state of the pupil. If you are using a white light for spotting the pupil will be shut down as in daylight. If you are using a red light the pupil will usually be wide open and may in fact be more problematic than using a white spot light. My personal approach to this issue is to observe the reaction of the bird closely for the slightest sign of stress or unusual response. If the flash had any adverse impact I would stop shooting immediately.