3D background for shoot'em up

Hi there, greetings!

I am back to game development after a while and I was wondering what would be the best way to create a 3D background effect like in Gradius V, so I could combine a smooth 2d scrolling with a 3D background that can turn in some directions. The resulting effect is pretty nice.

At first I thought about fixing the camera on the player so I could follow a path, but I realized it would be a nightmare to calculate all the required changes to position and rotation of all flying objects to keep the 2D look and feel of the game.

Anyone would like to help with some ideas?

Thank you in advance.

(PS.: I would not like to prerender the background as a movie and just play it on a surface, because this way I could not use the engine effects on it)