One is the reverse-direction (left-handed) drill bit, mentioned above. Two is a screw extractor, somewhat similar, but you drill the screw first with the correct-sized drill bit, then insert the screw extractor. Turning the extractor in the lefty-loosey direction, it goes deeper into the drilled hole and when it stops, the screw will (hopefully) unscrew from the hole. Three would be a manual-style impact driver. I would try this option first. You use the correct Apex screwdriver bit to fit the screw head exactly, Borrow the tools needed, if you can. An impact driver looks like a heavy screwdriver with a socket-wrench fitting, not a blade. Use socket adapters to install the correct Apex bit onto the impact driver end. Have an assistant hold something heavy (a brick or better) on the far side of the handle bracket. Press the impact driver/Apex bit into the screw head, turn the impact driver one click in the lefty-loosey direction, and hit the far end of the impact driver with a fair-sized hammer, to drive the bit deeper into the screw. The brick will prevent the bracket from turning, and hopefully, the screw will loosen part-way. Repeat if necessary, then use the impact driver as a common screwdriver to unscrew the damaged screw.
The impact driver can do the deed nicely, even with damaged Phillips screw heads. The other two options work best with Allen headed screws, but you can use them on Phillips heads as well. The better your Apex bits fit into the Phillips heads, the better the chances will be for success.
If you are buying the extractors, bring the correct screw (from the other handle maybe) to show the tool seller what size problem you have.
If all else fails, as a last resort, you can use a drill bit that is the same size as the screw threads to drill straight into the old screw head, just until the screw head comes off completely. Dis-assemble the handle bracket. Then unscrew the stub of the old screw from the handle using Vice Grip pliers or similar, and install a new screw.
Keep us posted,