If you want a low cost starting point, try a CCMT OD turning tool holder (SCLCR) and a CCGX insert (Shars.com is a good place to get these from). The larger the nose radius, the greater the required depth of cut and cutting forces, but the better the finish for a given feedrate. A small nose radius will provide tighter tolerance cutting due to reduced forces and allow you take a smaller depth of cut which might be a requirement for your lathe. See how long those last.
The CCGX has polished top face and a quite positive geometry. Because they are ground on the periphery, they are very sharp and tend to do well in soft gummy materials. Cutting oil or coolant are a must as the built up edge will quickly destroy the insert. The insert will cut well at any speed, however you will get the greatest productivity and tool life when running at the Mfg. suggested surface footage (aka. spindle speed). with such a small diameter, this recommended speed may be well beyond the capabilities of your lathe.
If you find that these insert wear out too quickly (not breakage, that's a different problem), then you might want to consider talking to a tooling specialist like Sandvik or Kennametal. They know their products and can recommend a tougher grade of insert probably with a coating to do the job you want. Expect to pay significantly more for this insert.