3 axis laser engraver

Hello.

I would like to trnasfer graphics from computer to round objects - <img src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQd_kN4025dQ9ABW3RGXfk6F-7tkS4y4p_mrFnI5kwan-gOdpc_aw">
I tried to use 2 axis laser engraver but it did not work well. In the center where the surface is flat it is ok, but on the edges where object get lower, laser loses its power...

So I am wondering, if it is possible to use 3 axis cnc router and mount laser head on it? Would this work? What kind of laser head do I need, it has to be just strong enough to burn down the colour. My plan is to spray the object with black spray, burn down the colour with laser to get graphics and then etch it in acid to get the relief out...

Thank you very much!