True enough: "partner" is ambiguous and I didn't detect the possible distinction. OP may or may not be regarded as married or equivalent for immigration purposes.

But the citation of EU rights remains the same: only difference is the loss of derivative Permit C status.

Under ECHR Art. 8(1) binding on Switzerland, the children have a right (not unequivocal but a right nonetheless) to be with their father. Likewise Chen/Zhu case (ECJ, but persuasive for interpreting the EU treaties). And OP has independent EU rights. Worry about the rest after he gets here.

And as for schooling: assuming they are Italian too where's the issue?

These aren't the bad old days: I remember Otti, nice guy, Italian, I knew him in Baar (Zug). Couldn't get divorced in Italy (no divorce in Italy in the 1960s, nor in Switzerland if the law of nationality wouldn't tolerate it) and couldn't live with his Swiss citizen girlfriend (hey, that would have been scandalous in those days). Worked for two decades in Switzerland: who would put up with that Victorian nonsense today?

Of course if they guy had been rich he could have gotten divorced in a third country. Nevada even. Like my Maltese friend.