I would create a custom shader that clips the wedge you want based on uniforms rather than using built in clip planes. This is similar to @prisoner849’s UV approach but it would not require that that UVs be set up radially which can be difficult with some models.
If you really want to use the three.js clip planes you can draw the model twice if you want to display > 180 degrees of the model. Once to draw the first 180 degrees and again to draw the remainder greater than 180. You might wind up with a seam at the joined edge but you can probably fix this by ensuring the two draws overlap a little bit.
Both approaches should still work with the stencil approach for the caps.