Keppel, is there any way I can have a look at your blend file? There's various ways to fix crossed edges caused by the Solidify modifier. In fact, I've dealt with a similar situation in my last tutorial, and discovered an easy fix to correcting that problem in most situations. I'd be interested to see if it works on your model.

The math function addon can be used to create a Cyclide in Blender, by inputting the formula directly, and then saving it as a preset. However, any torus primitive can be transformed into a cyclide, with some proportional modeling, or the application of some modifiers. The issue with that approach, in retrospect, is that it may not result in an accurate representation of the edge flow necessary for the mask, and extensive manual editing would be required. The result, after all, would be no more efficient than your approach. However, I do have a notion that this could be an excellent candidate for Bsurfaces, and spline modeling. I don't have time to try it, but I do suspect it may be a quick and easy strategy.

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