Is there a setting or some way to keep the HDMI1 input "active" and stay connected as a 2nd monitor when I change to another input?

Input Method Editor or IME allows you to switch between different keyboard languages in Windows 11 and Windows 10. It is available on the Taskbar and denoted by the selected language.

By dragging around the boxes here, you can have the digital display match how the monitors are set up on your desk. This way, you can seamlessly bounce back and forth between monitors.