There are a couple of ways to do this. You do not have to remove the FCG, and you want the bolt all the way to the rear, not halfway. That prevents the safety from falling down into the receiver.
The simplest is to put the shotgun in a padded vise, and use a good gunsmith driver with a sharp edge and simply bear down while you back the screw out. This doesn't always work. The center punch method helps if you have resistence sometimes. You can also tap the back of the driver with a hammer.
The next way is to put the shotgun in a vise, mask the receiver with heavy tape, and cut a slot in the existing screw head with a cutoff wheel on a Dremel. You can then back the screw out normally. If you do this make sure you have a replacement screw for the safety as it will destroy the one that is in there. If you bought the kit from Brownell's, it will have one.
Also, make sure you have the safety in the "ON" position to the rear with the bolt to the rear as well. This will keep the mechanism inside the reciever from falling out. Once you have the safety slider off, you can use a small amount of grease in the detent on the receiver to keep the ball from falling out.
Even with the metal safety sliders, it is a good idea to use the flat bearing plates under the safety slider. It will help minimize wear on the slider.
You may want to use some LocTite on the safety screw as well to make sure it doesn't back out. If you don't, just check it occassionally to make sure it stays tight.
If you get the BB stuck in the detent, good luck. I've heard all kinds of fixes for that, and there doesn't seem to be a universal fix for that. Cooling, heating, Dremeling, puch, gouge,everything's been tried.
Replacing the safety should take a few minutes, and no disassembly outside of the safety screw and slider. That somebody would charge $80 is outrageous.