AFAIK, there was a well-known D-WY poster by the name of "Wyodon" (I can never remember his actual, non-sock name though; something with an "H"). L-MS isn't really ridiculous/irrelevant either since there was a poster and party nominee on here with that avatar who ran a write-in campaign against Sessions in 2008.
Well, West Virginia has a somewhat substantial (relative to other states of course) green party; Wyoming had a Democratic governor who I don't remember the name of; the DC Republicans hold their meetings in a telephone box, but they still exist; and Mississippi Libertarians possess a Wikipedia page and a website.
I'd go for C-Hi, if only because Constitution avatars are, if nothing, inherently ridiculous.
Dave Freudenthal, who was only elected because Cheney screwed everything up for the Republican in the race.
And I agree on C-HI. It doesn't even have a Wikipedia page!