Your solution, thus-far, implies you are already calculating depth to determine the scale factor needed to simulate perspective. To make objects move "behind" the camera, test that same depth when you draw them and do not draw objects whose depth indicates they should be behind the camera.
if (object.sceneDepth >= 0) object.Draw();
I recommend going 3D, although I can't recommend any specific extensions. I would not avoid using 3D because of a perceived learning curve.
Doom and many other games use animated "billboards" to quickly approximate 3D. A billboard is simply a textured quad that always faces the camera. As the billboard rotates, the texture displayed on it can be animated to reflect the camera's new location and perspective. This allows you to render in "3D" while only actually rendering 4 vertices.
So, to answer your question, while you are able to render 2D sprites at different depths into the screen, it would be complicated, at best, to splice that together into something "like Doom".
If you look at the textures for Doom, you will find that every pose, for every creature was pre-rendered. When you move "behind a monster", the billboard is turned to face you, and the front-view monster texture is replaced with the rear-view monster texture. They were all 2D textures, drawn on 2D squares, oriented in 3D space.
Grass and trees are sometimes rendered with only one texture, but it is obvious because the trees can be easily noticed turning to face you. Their use is also evident in games where the grass becomes visible and appears to "stand up" when the camera gets close.