This seems to be a reference to to "Sansukumi-ken", an old version of rock-paper-scissors in Japan, where "slug" beats "snake", "snake" beats "frog", and "frog" beats "slug".

Nadeko is the "frog" here metaphorically to Kuchinawa's literal "snake".

I wonder if this metaphor was intended by the story, since Hachikuji ends up winning against Kuchinawa (and as a "snail" is essentially a "slug")

Shinjidude said:

I wonder if this metaphor was intended by the story, since Hachikuji ends up winning against Kuchinawa (and as a "snail" is essentially a "slug")

Yep. They directly reference Sansukumi in Owarimonogatari 2 when talking about their plans and they also mention Kaiki's Slug Tofu then.