Can BetterDisplay show a Dock icon while Picture in Picture is open?

Yes. Open Settings > Application > Notifications and Dock and set Show BetterDisplay in the macOS Dock to When a Window is Open. A Picture in Picture window then keeps the Dock icon visible even after you close Settings.

The default When Settings is Open follows the Settings window instead. Choose Always if you want the icon to remain available after all app windows close. These choices affect the app’s Dock presence. Window priority and the captured screen are configured separately in the Picture in Picture menu.

See Notifications and Dock for the visibility options and Picture in Picture for window priority and presentation.