If ths screw comes up a little bit, use a pair of locking pliers (vicegrips) to grab the head and pull up hard while turning it. This usually works on these pesky floor drain cover screws. Then you can use the next size up thread tap and put in a stainless steel screw (that way it won't corrode).

Note: I saw you mention using a drill to take the head of the screw off. You can do this, but you may need to use a pair of pliers to keep the screw from spinning while you drill it.