Can the Smirnov-Zamkow reaction be described as a pericyclic process?
I was going to write a comment, but I think this is good enough for an answer.

The issue that symmetry rules forbid a [2+2] thermal cycloaddition if the addition is suprafacial-suprafacial.

However, you can achieve a suprafacial-antarafacial [2+2] thermal cycloaddition. This is symmetry allowed because the antarafacial symmetry is exactly opposite suprafacial symmetry.

I'm not familiar with your exact example, but I am familiar with [2+2] thermal cycloadditions of ketene. It is thought to proceed via suprafacial-antarafacial additions.

See for example: Burke, L. A. J. Org. Chem., 1985, 50, 3149.