Just tested this. Simply Pre-compose the 3D layers and animate the pre-comp instead. You can animate it however you want, not only left-right.
Well, it's a 3D object in an environment with a camera...that's how it should look (perspective). You can fake the effect of it moving moving the background instead.
A workaround to what you need is to not make it a 3D in AE, make it a 3D in Illustrator and import it to AE without making it a 3D layer in AE. This way you have a flattened layer with a 3D aspect that will meet your requirements.
You can always go back to the AI you imported, change the text, save and Reload Footage in AE and it will update.
I am not sure if AE has a flatted or expand / expand appearance function to avoid switching between AE and AI.