MonoGame does not expose blitting functionality. The best you can do through the API is a draw call. A FrameBuffer in OpenGL is kind of the same as a RenderTarget2D in MonoGame.
So in MonoGame you’d have a RenderTarget with the result of rendering the mesh and you render that render target and the overlay stuff to the back buffer every frame. You draw the mesh to the render target when it needs to be redrawn.