Someone may actually make such a product, but just the veneer skin on the steel would not be a good idea at al. The veneer would be too easily damaged.
You could apply wood paneling to the door. That is done all the time. Can’t do it to a labeled door though, it voids the label.
I would suggest using a solid wood door of at least 2 inches thick to accomodate the cutout for the closer track, and make sure that you check the template for the LCN 2610 - it sits pretty close to the face of the frame.
I assumed a steel frame, but if memory serves me, the closer body still sits very close to the face of the frame. I have run wood trim over the face of the steel frame. I haver also installed this same closer in wood frames as well.