You could try a diluted white vinegar wash (1 part white vinegar to 1 part warm water). If it doesn't remove the rust, you'll want to take it a step further. Its a slow process, with having to "soak" the rusted spots for a good 6-8 hours, after its soaked for the required time, wipe it off with a lint free cloth and fresh hot water to get any residue off and dry completely
If that doesn't work, you could put undiluted white vinegar on a lint free cloth and cover the pickup with it for 4-6 hours, afterwards wipe off with hot water the same as above.
This will probably be the mildest of methods before you decide removing rust is more important than the finish, because any conventional rust removal product is going to ruin the finish. Obviously for both of these methods you'll want to take the pickup out of the guitar.