I believe the shim should have just brought the neck up relative to the bridge, to prevent the saddles on the BA bottoming out, right? If this is the case, then as long as you are not at maximum extension on the grub screws on the Wilkinson saddles, you shouldn't need to remove the shim [is it a full pocket shim by the way?].
I set-up my Geddy Lee & KSD 704 Jazzes with very little neck relief [and I think Bernie Goodfellow did too, but don't quote me] so they were almost straight and bizzarely enough, I found this helped with getting the action down. I then got individual string height / intonation sorted to get as low as possible without buzz using the saddle grub screws. I don't mind a bit of clank though!
It may help to give the rod a tweak and only use small [1/4-turn as you said] increments [righty-tighty - lefty-loosey]. Some advocate loosening first to check the rod's working before nipping it up. I'm not being patronising, but don't risk your neck if you are unsure. No idea what size it might be though, some f***ed up imperial size knowing Fender...
Good luck mush...