What is the actual combination of aberrations at play here,
The "spinning top" bokeh near the edges appears to be tangential astigmatism. Because the bokeh balls don't change their "width" (i.e., the diameter of the spinning top doesn't substantially change) further from the image center, the sagittal focus is fairly constant. But because the bokeh balls are getting squished the further out, that indicates tangential astigmatism—the tangential focus plane is changing as you move away from the optical center. This is also supported by the fact that in the first image, the linear lights are smeared along radial lines, increasing with distance from the optical center.
Normally astigmatism is best characterized when observing point sources at sharpest focus; in these cases, the lens is not focused on any of the lights. Hence, it's the deformation of the bokeh along tangential fan lines that gives away the tangential astigmatism.
What distinguishes this lens's astigmatism from the run-of-the-mill squishing of bokeh balls due to minor amounts of Petzval curvature is the radial smearing of the center of the soap bubble bokeh balls. (Note that astigmatism and Petzval curvature are related aberrations).
and what construction details should one look for when looking for a triplet with this kind of character?
I'm not ignoring this part of your question. I honestly have no idea what elements of the lens construction leads this aberration.