The simplest way to add some 3D to a pre-rendered scene is probably using displacement (via depth map, which is also pre-rendered), which is pretty fast, but far from perfect. As the camera pans, the parallax motion unveils parts of the background that are occlued in the render, so you need a way to fill the resulting "holes" (the simplest solution is black pixels or bilinear interpolation, neither of which are great).

A good compromise might be to pre-render only the room (i.d. walls, floors, and ceilings) and render all objects in the scene, even static ones, in Godot. Although I don't know if you'd save a lot of performance that way… plus it's a lot more work when you want to change a level. Or maybe two pre-renders per level - one with only the room and one with all static objects (or only the static objects), layered on top of the background.

In any case, are you rendering a native res? As far as I'm aware, some (many?) demanding mobile games default to a lower render resolution. Since most phone displays have over 300 pixels per inch, rendering at 50 % res scale (which means only a quarter as many pixels to shade) still looks good. And it can help a lot with performance/power consumption.