The best complex dielectric mirrors, see
may reflect up to 99.999 percent of the incident energy. The loss is about 1/100,000, so after 100,000 reflections, the total intensity decreases $e=2.718$ times or so. If the distance between the mirrors is 3 meters, the light travels 3 meters times 100,000 = 300,000 meters before it gets diluted $e$ times, and it takes 1/1,000 of a second. So within one millisecond, most of the light is absorbed, anyway.
The time may be extended by increasing the distance between the (great) mirrors.
In principle, if the mirrors got better, a photon could be trapped. Its phase would be changing by the gravitational field – the photon would literally start to accelerate downwards. With the realistic mirrors described above, this acceleration downwards is pretty much unobservable.
More generally, the absorption by the inevitably imperfect mirrors is still the fastest process that makes the vision about the trapped photon impossible. Within the time scale before the light gets absorbed, the photon may be considered almost perfectly trapped because all other effects that violate it are negligible in comparison.