I would have been delighted if some Forum members had already voted on this, but since the question has remained unanswered for over 12 hours now, I’d like to offer the following observations:
The sentence “A cat with two kittens is sleeping” is both notionally and grammatically correct. This means there's no conflict between notion and grammar at all. The subject of the sentence is the singular noun “cat,” so the singular verb form “is” agrees with it. The phrase “with two kittens” doesn’t make the subject plural because the preposition “with” only indicates combination or accompaniment, not the compounding of the subject into plural form. This is why the sentence "A cat with two kittens are sleeping" is grammatically incorrect.
What would require the plural verb form “are” in that sentence is the additive conjunction “and,” as in “A cat and its two kittens are sleeping.” We also must keep in mind that even the use of the prepositions “along” or “as well as” won’t require a plural form for the verb; the verb will remain in the singular form, as in “A cat along with its two kittens is sleeping” and “A cat as well as its two kittens is sleeping.” (Strictly speaking, though, the preposition “as well as" needs to be set off by commas to avoid confusion: “A cat, as well as its kittens, is sleeping.”)
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