Chris, I am not a clockmaker but I do use a Sherline lathe and I like it. You are working with really small diameters in some rather hard steels so your tools will matter.
I think the biggest issue might be that you're using brazed carbide. I say might because the tip geometry of a brazed carbide tool generates a lot of cutting forces if left in an unmodified condition. Rather than jumping to conclusions, can you either show us your tools or describe them? What kind of nose radius do they have? Have you changed the original tool geometry at all?
I ask because you're working with tool steels and they can work harden if your tool is not cutting continuously. If the work hardens then it is harder to bring it to precise dimensions and will also make it more likely to snap off if excessive tool pressure is used. Brazed carbide tools will work for you but they typically have fairly shallow relief angles and little, if any, rake. They are also not typically sharp as received. You can modify them with a green silicone carbide wheel or a diamond wheel and they will work better for you, provided you know what you're doing.
A better option would be to use HSS lathe tools or gravers. I say better because HSS tools are easily ground on a bench grinder or belt sander that is more likely available to you. If I had to cut small diameters like you're shooting for, I would use a cobalt HSS knife tool with a 1/64" nose radius, 15 degree relief angles on the side and end, 15 degrees of side rake and about 12-15 degrees of back rake. I would hone it with diamond stones to get rid of grinding marks, then use a fine India stone to hone it well. Finally, I would use a Translucent Arkansas stone to polish it. A tool that is ground and honed like this will cut small diameters readily and will take cuts in the tenths. It is not good for stock reduction though; a general purpose tool would work better for that. So, I would rough it with a general purpose tool and then switch to a knife tool to get it down to size. I would use cutting oil for all cuts.
Gravers will also readily cut your small diameters. In experienced hands, gravers can turn very precisely and will work well for one-off parts.
Your lathe will work for this, Chris. Just use the right tools and you'll do fine.