I dont know of a single shop where if you were a project manager and insisted on making that in a lathe you would wouldn't be fired. Waste of time, lathe dosen't have natural work holding so you would have to make one, lathe would need live tooling and long reach Y access ability. If you tied up a lathe with the capability to do this you'd be losing on all the jobs you'd miss while you messed around.
First line on your project should say "your fired" then explane the reasones why it shouldn't be done on a lathe, and finally do it for a mill.
In the real world a machinist would assume it was a wrong checked box to use a lathe and would have made it on an available mill.