...Completely stumped
First, new strings are more likely to buzz, not less. No denting of the wrapped wire means metal closer to fret tops. I think OP said his buzzing went away after a while of playing.
You should also give some additional details like how old the bass is, when was the last time it had no problems, etc. But approach it logically (I'm taking you at your word that you've done every other standard setup troubshooting thing -- and English major note: it's gamut, not gambit

):
-- It's unlikely every single fret has the same exact problem only under the E string (like all of the frets have heaved evenly). So it's not likely a fret problem.
-- If none of the other strings have any problem, and assuming all four strings are set to the same action (following the radius of the neck), it's not the neck.
-- Are you 100% sure the buzzing is from the string hitting the next highest fret every time? take a very thin piece of fabric and put it over the next highest fret as you play each fret -- if the buzz remains unchanged, then you probably DO just have a high/heaved fret or two. And also put physical mutes on the other strings just to be sure.
-- when you're testing, use a pick and don't dig in -- pick light but clear notes and see if that still buzzes -- reduce the amount of string motion. If you play with a heavy attack, you might just need higher action and more relief, or get comfortable with a bit of E buzz (that will die down as the string breaks in).
Worth a try?