Yes, as noted, I have a feeling you're better off using relays (or triacs!) and an alternate control scheme. 4A seems to be a lot of current for a rotary switch; you're asking to put almost 1kW through it. Sure, for "simple" variants (e.g. 1-deck, 3-position) you might find those and they might be reasonably priced. Otherwise, the only option I'm even seeing is NKK Switches HS16-4N which is going to set you back >$70.
Honestly, your lowest-BOM option might be to use a pot and an MCU... at which point you might consider adding a thermistor/thermocouple so you can actually control the temperature.