Can Mission Control affect only my virtual screen?

A BetterDisplay virtual screen is another display in the same macOS session. It does not provide a separate session with its own isolated Mission Control. Showing that desktop in Picture in Picture does not change this distinction.

If your aim is to keep the Picture in Picture window visible while you use Mission Control or change Spaces, open the source display's Picture in Picture menu and choose Above Menu, Dock, Spaces. This window-priority option places the picture above those system surfaces and across Spaces. Picture in Picture requires Pro and the applicable Screen Recording permission.

This changes where the viewing window is shown. It does not confine system-wide actions to the virtual desktop or isolate the apps running there. Choose Normal Window Priority or Keep Above Other Windows when you want the window to participate in ordinary desktop presentation again.

See Picture in Picture window priority and virtual screens, mirroring and streaming for the separate roles of the desktop and its viewing window.