I struggle to define this use of 'arch'; i would say it applies to speech or writing which says something indirectly, in a clever or playful way. Wiktionary says "knowing, clever, mischievous", and has some quotations [1].

I agree with Butler-Gallie here (not words i thought i would ever write) that there is nothing particularly arch about "shat".

[1] https://en.wiktionary.org/wiki/arch#Etymology_2