I have seen a non CNC version in the past where the router was attached to the tabletop and followed a different type of template.
What you are wanting to do is feasible. But you will probably have to right your own Gcode though. This should not be too difficult since you will only be telling the X and Z axis what to do. There is no Y axis with this specific setup. On q typical CNC lathe the spindle head of the lathe would be the Y axis, but it might be considered I,J, or K, I can’t remember exactly bc I have no actual experience with a CNC lathe. I am sure others here can correct m if I am wrong. But it really does not matter in your situation. The Lathe spins at it’s designated speed and that is it, it won’t reverse or index (unless you manually perform this). In your situation you will turn the lathe on and get it up to speed prior to initiating the CNC functions. So to execute your plans the X and Z axis are the only ones which need to move.
Also keep in mind how the cutter head will attack the workpiece. You will probably need a ball end mill
Another individual on the forum here has set their lathe up with a motor on the front of the machine, “The Mysterious 4th Axis”. This is similar to your setup but his has more control of the spinning axis by his computer software. You might look for it, he can probably give you decent info on how the code you will need.
Good luck