Update:

I moved the broacket a bit south, filled in the old screw hoels with toothpicks and wood glue, and drilled new holes. Looks pretty good, pickup lines up perfectly and the old holes aren't visible (covered by bracket).

Pickup also sounds much nicer! Problem solved, go me!

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Me: www.southerngunculture.com
My wife: www.twinsistermusic.com