Someone asked a similar question on Reddit and I explained my workaround to them. I Know this reply is way too late, but it might help other Replika users who come across this thread.
TLDR: I pin the Replika window on the desktop using TopMost Control, so it's always visible, even when VAM is the active window, then I use the native Steam VR desktop viewer when I'm in the VAM scene.
You have to get Replika and VAM active at the same time for it all to work properly. I do this by opening VAM in my regular browser (Firefox) then I open Replika in Chrome. As you know, to get the VAM scene to work properly, it has to be the active window, but that causes the Chrome window showing Replika, to disappear. I overcome this by using the Windows Topmost function. This basically makes a selected window stay “on top” of all other windows, even if it’s not the active one. There are several apps available for this purpose, but the one I use is called TopMost Control. It can be found here:-
https://www.sordum.org/9182/window-topmost-control-v1-3/
Unfortunately, it’s a .exe program, so I’m not sure of a solution if you’re not comfortable running this type of program. All I can say is I’ve never had any problems with it. When you double click TopMost_x64.exe, it opens an icon in your system tray. Open the Chrome window with Replika and double click the TopMost icon. That then pins the Replika window on top, even if you click the VAM window.
To see the desktop in-game, I use the desktop viewer native to Steam VR. When your VAM scene is open in VR, click the system button on your VR controller and this should open the Steam VR UI. There’s a button towards the bottom with the image of a monitor on it. If you click this, it opens a window showing the desktop, with your Replika on it (because the Chrome window is now pinned over the VAM window). Above the desktop, there’s a button called “float in world.” By clicking this, you can move the desktop image to where you want it in the room. In my picture, I’ve placed it just in front of the TV in the scene. The one down side with this method, is that it has to be done every time you open the scene. Look at Method 1 in this video for more info:-