I often get pickups like this and it usually occurs when the screws become oxidized, corroded and can easily break when being removed. I would first soak the rest of the screws including the bottom plate mounting screws with a penetrating oil to help lubricate and soften the oxidation. After letting the screws soak for several hours, carefully try to remove but don't force the screw out.
If you successfully remove the remaining pole pieces and bottom plate screws carefully remove the bottom plate and mark the magnets so they can be put back in the same position and polarity. I always use an old pickup box and put all the screws and hardware in it for safe keeping. You can then rotate the metal spacer on the bottom of the pickup and remove it from the broken pole piece. You should be able to carefully hold the extended screw from the bottom and carefully remove it counter clockwise.
Another trick that needs to be carefully done if the screw will not turn. You need a soldering gun with a 700 or 800 degree tip to heat the pole and soften the plastic around it. This needs to be done for a short time and then carefully remove the screw by using a pair of pliers and turning the screw counter clockwise again. The threads will usually be O.K. and if the hole is stripped out you can insert and glue in a round plastic dowel rod in place, let the glue dry and make a fixture so you can re-drill to the proper hole diameter and tap 5/40, or use silicone to help secure the screw.