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I have to give credit to my local radio board for this one. It quotes from Eric Alper, a well known local musicologist, noting that the Beach Boys actually crooned a tune for an obscure 1964 NBC TV show about a rambunctious teenager. The show, "Karen," ran on NBC for exactly one season and you've probably never heard of it.
Or the theme song, which you can listen to below.
1960s Sitcom “Karen” Captures SoCal Vibes with Beach Boys Theme Song
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This show clearly looks like an attempt to copy the teenage-girl-and-her-sister format of The Patty Duke Show on ABC, which had debuted the year before (1963). Alas, it was not to be. Karen lasted just the one season, while TPDS was on for three (1963 - 1966).
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The Theme of The Courtship of Eddie's Father (1969-1972) was written and performed by Harry Nilsson.
(I love the 30-second introduction that precedes the song)
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I don't think the Beach Boys ever had a commercial release of the Karen theme, but the Surfaris did in 1964.