Say we have our reflective container, 1 meter on a side (cube shaped). The solar constant is 1366 Watts / meter^2, so our container will receive approximately S = 1366 Watts from the sun. The amount of time a photon remains inside the container (entering, then bouncing off the reflective surface) is T = 2 meters / 3x10^8 meters/sec = 6.7x10^-9 seconds.
The number of photons trapped in our container if we close it infinitely fast is N = S*T/E, where E is the energy of each photon. For simplicity, assume all the energy is in visible photons, of wavelength 550 nanometers (yellow light), which of course is not the case.
The energy of such a photon is E = h*c/wavelength, with h = planck's constant = 6.6x10^-34 Joules*seconds, c = speed of light = 3x10^8 meters/second, and wavelength = 550x10^-9 meters. So a single photon has an energy of 3.6x10^-19 Joules, and the number of photons trapped is an impressive 2.5x10^13.
Say our container, having trapped this many photons, is 99% reflective. Thus, every reflection dissipates in some way 1% of the existing photons. Because our container is 1 meter deep, we have 3x10^8 reflections every second.
This exponential decay could be represented by N = N_0 * e^(-t/tau), where N is the final number of photons, N_0 is the initial number, t is the time between capture and release of the photons, and tau is the "half-life" of the photons for getting dissipated instead of reflected. A single reflection takes 3.3x10^-9 seconds and the final number N = 0.99*N_0. Using these numbers, we find that the "half-life" tau = 3.3x10^-7 seconds.
Now, we have trapped our 2.5x10^13 photons. If we release them after one second, we'll have N = 2.5x10^13 * e^(-3x10^6) = something essentially equal to 0 (far far less than 1 photon remaining, on average).
Fiber optics are about 99.99% reflective, but even this level of reflectivity would give no remaining photons after just one second. Photons traveling through very good fiber optic wire still dissipate at a rate of 10% per kilometer, which distance the light could cover in 3.3x10-6 seconds.
So, funneling the light through fiber optic cable can work, because light travels so fast, but carrying it in a reflective container is a losing game.