Discussed this today with the English Dept. where I teach, no definitive answer.
Is it “shat?” ex. The dog shat on the rug this morning.
Is it “shitted?” ex. The ferret shitted under the entertainment center this morning.
Is it “shit?” ex. The baby shit in his pants this morning.
I really don’t know the answer to your question, but it sounds like you had a hell of a morning!
My sense is that it is conjugated in the same way as the word “hit.”
So past tense would be “shit.”
I’ve always preferred “shat.” I just like the way it sounds.
According to Merriam Webster, both “shit” and “shat” are accepted past tense forms of “shit”. (You can also click to hear the pronunciation if you wanna amuse yerself at the office…)
It certainly is ‘shat’ in my part of the English-speaking world (England).
I was shat upon by birds* just the other day, BTW - supposed to be lucky; didn’t feel like it.
*[sup]gulls, no less - AKA ‘shitehawks’[/sup]
I go with shat as the past tense. Luckily we aren’t Roman or we would have
shiti,shitisti,shitit,shitimus,shitistis,shiterunt for the past perfect.
Shat in my part of England too. Isn’t shite runt a great insult though?
I believe Shitius was emporer Commodus’ older brother.
I always thought shat was a humorous construction based on spit/spat because they’re both bodily functions. Being as though it’s somewhat slangy and vulgar, I wouldn’t have thought there was a clear answer to this but Merriam seems to think so, and so does Webster’s Ninth New Collegiate. In reading the notes, there seems to be no etymological basis for shat as the past tense, although that don’t mean. . . .you know what.
I prefer shat, as well.
I wonder how William Shatner feels about this.
sit/sat?
What’s wrong with “took a shit”?
For me it’s, “Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh!”
According to Webster’s Third, the correct past tense form of “to shit” is “shit.” “Shat” is merely a variation (and a horribly sounding one at that). “Shite” is obsolete. Typical British.
Comparing to other words is no good. For “sit” and “spit”, you also have “hit” (hat?) and for that matter, “fit” and “slit.” It’s time like this I wish I had an OED to give me the long explanation, no doubt rooted in the varied declensions of Old English nouns.
Aye, but ye dinna take a shit did ye? Ye left a shit, ye shit. 
Ive always been curious about the name “shitehawk” given by beleagered British picnic-goers to an especially aggressive breed of bird encountered in India, my having grown up in the American Midwest on a river poplulated by a species of crane called a “shitpoke,” after its ballast-dropping behavior when frightened into flight.
Usta be that for a word to be accepted into standard usage it was proposed to committee and inspected and voted upon, etc.
Nowadays whenever something comes into common usage a new dictionary is issued. It’s good for business.
When and where I grew up “shit” was not even a word, but a vulgar colloquialism. It was also used as a noun, to describe feces. It was used with the verb “to take” much in the way that we “take” a rest, a bath, a powder, a ride, etc.
Until recently there was officially no such verb, “to shit.” The true verb to describe the elimination function referenced was “to defecate,” and the past tense was ‘defecated.’
(I thought bathroom humor was left behind upon entering high school… ?)
Strangely, I would say:
“The dog shit on the floor this morning.”
but,
“We didn’t talk about anything important. We just bullshat.”