Most dual tuner Sat boxes come with at least 1 RF remote, and they do output to analog channels, so what you are describing should be possible without spending any more money than the number of tuners you want.
So if you can live with 2 tuners, you only need 1 dual tuner box which should come with at least 1 RF remote and I'm sure extras are cheap on eBay if you want them.
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