I would remove the other screw to remove the bobbin making it easier to work with. If the screw shaft is still extended from the bobbin, you might be able to grip the shaft and turn it counter-clockwise. Carefully try to remove it without doing further damage to the bobbin. At times when the screw is put in, it become hot with friction and the plastic becomes tempered or hardened. I would suggest heating the screw shaft with the tip of a soldering iron to soften the plastic around it and then carefully twist and remove the broken screw. If this doesn't work, heat the screw shaft with the soldering iron and from the other side of the bobbin, use a paper clip to push the broken screw out. This usually works well.
Always use eye protection and perhaps gloves in case you touch the hot screw or soldering iron. Like I always say, if you don't have the proper tools, please send it to a qualified repairman.
WRITTEN ON JUNE 12, 2015, BY