I don't know much about bass, but I would agree with johnny7, go with heavier strings. Raise the action a hair and everything else should be ok, but intonation will probably need to be reset. I doubt if a new nut will be needed, string height at the 1st few frets shouldn't be an issue. Vibration there is minimal, if it doesn't have buzzing problems now, changing tuning and heavier strings shouldn't cause buzzing, which is the main reason you would have to replace the nut, to raise the strings a bit in that area.
Truss rod shouldn't be an issue, the only time you need to touch it is if it has buzzing problems around the 8th - 10th fret area, which would mean it needs more back bow. Ignore anyone who says the truss rod has anything to do with action or intonation, that's a myth perpetrated by the uninformed. The truss rod has one function and one only. To increase or decrease back bow to allow for the greater vibration of the strings in the middle of the neck. If you want a more detailed description, I'll be glad to do some more typing...
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