Sometimes if we don't have the most cromulent word, we have to make one up.
I think the word squee is starting to catch on. Squeeness actually brings up a few google hits.
Not in the dictionary, yet. Squee's attested by Oxford Living Dictionaries website now:
EXCLAMATION
informal
Used to express great delight or excitement.
…
Check out This Kitty! Squee!
I would call the word itself a pet term. This is an interesting topic, and pet terms are probably common within families and other small, tight-knit groups.
An example of this usage, in a headline from a media analysis website:
Limbaugh Explains His Pet Term "New Castrati": Men Who Are "Bullied By Women And The Power Structure And Liberalism"
Note that this is not the same as a term of endearment.
EDIT: This is not an established linguistic term. I consulted a few sociolinguistics textbooks and didn’t find any discussion of this kind of ephemeral in-group language. One place to look would be studies of college slang, like Connie Eble’s Slang and Sociability.
Pet term is also different from pet word, which seems to have a generally accepted meaning: a word that is frequently used or otherwise favored by an individual person or writer. Pet words are already part of the general lexicon, though. Brad Leithauser has a 2013 New Yorker piece about pet words. He gives some examples (sweet for Shakespeare, lad for A.E. Housman) and compares them to stray cats taken in by their users:
Each of these words presents the critic with a little puzzle of devotion: What was it about this particular package of syllables? Why was this stray cat escorted into the author’s studio and offered a saucer of cream and a plump pillow by the fireplace? It’s not as though the studio were soundproof; during working hours, the author no doubt could hear other strays, seemingly no less deserving, meowing clamorously for admission.
Link: Pet Words
So, pet term (rather than word) can convey that it's favored by the in-group, but not established as a true word yet.
Best Answer
While I think your suggested chagrin may be most on point, you also might use
In a sense, D'oh is a sort of onomatopoetic representation of the slapping of the forehead while exclaiming, I feel like a dope!