Hi,
I would guess you could do the same design that you were thinking of, but use transmission gates instead of relays. Do a search for 'transmission gate'. There are other ways to tackle it, but I think this would match your original idea very closely.
You could have a look at
this chip. It has 4 transmission gates built in. There should be a number of implementations, each of a '4066' designation. The switch resistance is ~80 Ohms. I think most relays will give much lower resistance than this, though.
I seem to recall there are also solid state relays available now (I've only used the electromechanical types). I think internally, they are similar to photoconductors or MOS transistors. These should be smaller, less noisy, and maybe have multi packaged designs, and presumably low activation currents.
Otherwise, you could have a look inside the remote control, and see what kind of signal is required internally. Then, design the transistors to affect the signals as required. This might be more of a hassle for you though.
edit- I haven't looked at this very closely, so take all the usual precautions.