Sometimes is beneficial to "freeze" an application, for example when configuring on-screen controls for a game which doesn't display them at all times. Hotkey would be the most convenient.
What OS are you using? I'm going to assume Windows. I have no real experience with Bluestacks itself, but you can try suspending the whole process. To test if it can work open the
Resource Monitor, right-click the Bluestacks process and click
pssuspend <process name | process id> (
-r to resume)