Maybe! Actually your estimate of one photon out of billions is wildly optimistic; the best lunar ranging experiment that's been built (the Apache Point Observatory Lunar Laser-ranging Operation, APOLLO) sends out 300 quadrillion photons per pulse (115 mJ/pulse of a 532nm laser) and can detect an average of 3 return photons per pulse if it's pointed at the Apollo 15 reflector (the biggest, and therefore brightest, one available). That's one photon in 10^16 actually makes it back — less if using one of the smaller reflectors, and less if the conditions (position of the moon, atmospheric stability, etc.) aren't perfect. And yet, that rate corresponds to 3,000-4,000 reflected photons detected per minute, which is a huge improvement over previous instruments.
Which gets to my point — lunar ranging experiments started 50 years ago, when the technology wasn't as advanced as it was today. Back then, they were lucky to get one photon per minute of genuine signal, but it was still enough to do science with, if left running for a while.
So is it plausible that a rooftop experiment could mange to be 1/4000th as good as the Apache Point operation? Well, probably not. The telescope should be bigger, and the location should be away from the city, up in the mountains, and not on top of a building. Cost-wise, you would need the telescope, a pulse laser (only one watt or so of average power, but compressed into brief, intense pulses to make the signal stand out above the noise; a regular laser pointer won't do), an avalanche photodiode detector, a mount capable of aiming the laser and the telescope to arcsecond accuracy, a good time-to-digital converter, a bunch of slightly-exotic electronics, and a computer or two. But you know what? I think it might actually be doable on a roughly $5,000 budget by someone with the right set of engineering skills and a lot of dedication and time to put into the project. For TV purposes, we can handwave some of these things away if needed. But I'd love to see someone replicate this in real life!