These problems are a bit easier to think through if you're careful to name them in terms of what you've got, not old-fashioned terms you're used to using.

You're not trying to individually control the channel on each TV in the basement. You're trying to individually control the channel on each DirecTV receiver in the basement. In the first case you'd be looking at TV remotes; you want to look at DirecTV remotes.

The RF mode of DirecTV remotes couples ONE remote with only ONE DirecTV receiver using part of the receiver number in the setup procedure. This should mean you can control a hundred different DirecTV receivers, each with its own RF remote, right next to one another. (You also can set up a total of 8 different IR codes on DirecTV receivers, even setting as many as three per remote, using the slide switch to choose.)(You can also set up more than one remote to control one receiver via RF, but not the other way around.)

A DirecTV receiver set to RF will not respond to IR. I believe this might not be true for the Genie and its little Geniettes, but I'm not sure.

I don't know for sure about Charter cable boxes, but I have yet to see a cable box that could use a codeset different from another box of the same company. I also don't know about Dish network product. The client is asking there for something outside his budget.