how is everything? I have a question and I hope to find an answer. As you already know, there are four types of second-inversion chords: cadential, passing, auxiliary, and bass arpeggiation. My question is, does anyone know of any second unclassifiable inversion chord in classicism? I need an example of a piece where this happens
We typically classify these chords by what their bass notes do. Thus a passing six-four is so named because the bass functions as a passing note; the same is true for the arpeggiating six-four.
Your auxiliary six-four is also often called a "pedal" six-four, and once again that term describes the bass. The cadential six-four, a subset of this pedal six-four, follows this same rule of being labeled by its bass motion.
So we can label any six-four chord we find just by labeling what its bass does. For example:
In the above example, beat 3 would be understood as a "neighbor" six-four chord, because the bass is a neighbor tone to the surrounding Es.
And in this example, we could call the second chord an "incomplete neighbor" six-four chord. This is like a neighbor, but instead of the bass being a step away from the notes on either side of it, one of these notes is farther than a step.
But even though we can label both of these chords by their bass notes, remember that they are not stylistically correct in the Classical style. The reason we teach those four types of six-four chords is because those are the types used in this style. Any other type of six-four chord, no matter if we can label it or not, is to be viewed as an "illegal" type of six-four chord in this style.
The logic is that, historically, the fourth was viewed as a dissonance. As such, a six-four chord would have this dissonant fourth between the bass and an upper voice. In order to soften this dissonance, the six-four chord could only be used in particular environments: as a passing, pedal, cadential, or arpeggiating six-four chord. Any six-four chord that is not one of these four types is understood as being outside the Classical style.
Note that some composers like Mendelssohn, Wagner, and Brahms began using these "illegal" six-four chord types more commonly. But they were also composing in a different style; these chords are somewhat "legal" in Romantic music, but not in the Classical style.
Although I'm not sure if these examples will fit your stipulation for "classicism," here are two instances of "illegal" six-four chords in the repertoire.
Above is from mm. 69–70 of Bach's "Wachet auf" (Cantata 140). Note the illegal "neighbor" six-four chord on beat 2 of the second measure.
And here's another "neighbor" six-four chord from Mendelssohn's Op. 19, No. 4, mm. 19–20.
These two examples came from David Temperley's article "The Six-Four as Tonic Harmony, Tonal Emissary, and Structural Cue." In that article he also has examples by Schubert, Schumann, Chopin, and several others.
Edit: On second thought, I'm unhappy with both of these above examples. The Mendelssohn could be viewed as a cadential six-four, as Jose says in the comments, but its resolution to a Bm7 prolongation is still a bit strange. (Perhaps it's a global passing six-four chord now?) And the Bach could be viewed as a hybrid cadential/arpeggiating six-four chord. I'll leave these up for now, but I'll try to remember this answer and provide better examples if I encounter any.