How close (vertically) can I safely relocate a lag screw in a stud?
It really depends on your technique. The stud doesn't care. Plenty of them have knots bigger. If you can leave even half the hole diameter between the new and old holes it'll be virtually as strong as otherwise.

Mount the bracket using other holes, then drill the new hole. The steel will prevent your bit from dropping into the original hole. 

Of course, you can epoxy a dowel (even hand-carved from scrap) into the hole and start over at any location.