A quick shot at Google using "multi-monitor kvm" search found a company with the features you've noted. Good thing your budget is flexible.
Dual monitor systems appear to be much more economical than triple (in progress, per one site) or quad.
KVM Switches Online does a great job of categorizing the products available, specifically in this case, allowing you to pick HDMI for dual or quad monitors. One of the switches supports USB 3.0 while the other list as USB 2.0 only. The USB 3.0 unit is listed as a hub, but the image of the front shows 3 ports. The rear of the unit has 2 USB ports marked for keyboard and mouse.
I should read your question before I hit the submit button. The device listed supports 4k video resolution as well as audio switching. The information leads me to believe that the switched-off unit gets muted, which is as it should be.
A reference is made in the PDF for the product of hot-key switching but no specifics are available.
As is typical of these products, cables are purchased separately. It appears that these units do not require proprietary cables.
The Q/A section of the site suggests to a user that if the above item is inappropriate, to purchase a different (USB 2.0 version) instead. It appears to be nearly the same cost, but with more USB ports on the panel.