How can I switch multiple circuits using switches with limited pole counts?
Search for wafer switches. 

They are rotary switches, which used to be available (update : still are!) as pre-built configurations, or in modular form with up to 12 ways on each wafer and as many wafers as you felt like stacking behind the switch body. 

A 4-pole 3-way switch is a single wafer, one of the simpler standard configurations.