If I met a family whose kids' names all started with J, I would immediately have the suspicion that they're right-wing evangelicals like the Duggars, and would be cautious about closer acquaintance; I may be an outlier in that assumption, though.

Is there any possibility of kid #5? If there is, that's another reason why I'd go for a non-J name for #4; otherwise, you're really stuck with a J for #5.