The word "ばかり" in Japanese is not an adjective (な-adjective or い-adjective) but a particle or auxiliary that is often used to express the meaning of "just" or "only." It's used to indicate that something has just occurred or is limited to something specific.
In your examples:
海外で一人暮らしを始めたばかりなので、寂しくてしかたがありません。
(Because I have just started living alone overseas, I am so lonely.)
子どもを産んだばかりなので、うちで休んでいるところです。
(Because I have just given birth to a child, I am resting at home.)
In both sentences, "ばかり" is attached to the past tense of a verb (始めた, 産んだ) to indicate that the action of starting to live alone or giving birth has just been completed. The structure [verb in past tense]+ばかり is then followed by "なので" to provide a reason or cause, consistent with the usage of "なので" after nouns and な-adjectives.
So, "ばかり" is not categorized as a な-adjective; rather, it functions as a particle or auxiliary that modifies a verb.
Hope it helps.
I wonder if maybe there is in fact something "nounlike" about ばかり
Does this mean you are looking for the 'noun-ish' feature of ばかり so that it fits well with なので? Although you saidthe assertion that なので can only be used with a noun or a な adjective is simply an overgeneralization.
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