If your reception is already marginal than there's not much needed to stop it from working. First thing I'd say is trying to get a better reception by either moving the receiver to a better spot or hooking up an external antenna if that is possible. Of course you can also try to find the course of the interference but that will only work until you bring in another device and kill the reception again.
Do you have a spectrum analyzer? If so you could examine the clock with a (home-made) near field probe and see where the problem is. If you stop the multiplexer is the problem gone? How do you create the 5V AC, is there an oscillator involved that is a bit too enthusiastic? Switch it off and see what happens. Ferrite beeds can actually help to suppress unwanted radiation, try them on the 12V entry and make sure you have no wires longer than needed. Lastly, a picture would be nice!