For much of this kind of work, round head screws and such, I use a plain lathe bit, but move the carriage toward the headstock and crossfeed out at the same time to replicate a curve. An easier approach is to move the carriage in a little, move the cross slide out a little, step and repeat until you have a shape close to what works, then finish the shape with files. With a little practice, you will get quite good at turning a radius on the end of a rod.
A lathe bit ground with a small rounded nose will nicely between the ball of the screw where it necks down to the collar of the screw.
As in all metal and woodwork, knowing the job you need to do often requires that you shape a special tool just to do the job. Eventually you will end up with a drawer full of tool bits of different shapes and sizes, from thin blades, to vee tools for threading, to convex nose tools, to concave edged bits. The list is almost endless.