It might be possible to adjust their approach somewhat and detect the reflectors with a group of smaller telescopes (at the high end of the amateur range). Their detection technique involves splitting the light up among multiple detectors anyway (allowing them to time multiple photons individually), so this might actually be a way of increasing performance.
Also, their laser is fairly low powered, just a couple watts, but designed to produce very consistent 100 picosecond pulses. You should be able to get by with longer pulses with a much higher power laser, especially if you just want to detect the reflectors.
It'd still be very expensive and of doubtful scientific value, though.