Not a mugwump; a mugwug.
The first hit I get on google is a Facebook page that says ‘A mugwug is a person whose education exceeds their intellect.’ Aside from that, I’m just seeing it appear as a bunch of usernames.
It must be a pretty recent neologism if Urban Dictionary hasn’t heard of it.
Prolly a first cousin to a pollywog and best friends with a wallypug.
Sure it wasn’t meant to be mugwump? Old political insult. (I was first told its late, evolved meaning: a fence-straddler with his mug on one side and his wump on the other. The original meaning is a lot closer to what the OP thinks it is - determinedly stupid pols.)
Google N-gram viewer gives zero hits when you type in “mugwug”, which is weirdm, because Google Books gives you several hits, many of them before 20000 (the N-gram cutoff date), including one from 1937.
All the uses sem to be the same as for “Mugwump”, so this might be a case of people reverting to “rhyming variation”, in which they tend to recast words as rhyming halves*. In any event, “mugwug” seems to be used synonymousl;y with “mugwump” in these older uses.
Here’s the 1937 one:
*the way hara-kiri became hari kari in the US in the 1940s-1960s. Helped, no doubt, by Harry Carey’s name).
John, it’s Facebook. People, even smart people make mistakes. It was either a typo or the person just didn’t know how to spell it.
Remember Occam’s Razor. Or K.I.S.S.
I only found the FB page because I searched to find out what ‘mugwug’ is. I actually saw it on a license plate.
You do know that the most common original use of “vanity” plates was to put your initials on them - typically husband’s and wife’s? Making two three-letter groups, often with the same last letter? It might be a bit ol’-fashion, but seeing MUG WUG would make me think that before trying to assign a meaning to it.
Typically older Buick-y cars with an older driver or an older couple, in my recent experience.
BTW, John, I really wasn’t calling you stupid. 
I hope you know that.
Was the license plate the old style? They only allowd 6 letters, so, maybe that was the closest they could get to what they wanted.
New plates (which are replaced every seven years). I believe she intentionally chose MUGWUG, rather than trying for MUGWUMP, which would have been MUGWMP or MGWUMP.
I know this person, but I don’t want to talk to her. No apparent spouse.