How to replace difficult outside hose spigot handle?
If you haven't stripped the screw head, they sell impact screw drivers for just such a situation. Like this one.

A tool like this allows you to strike the impact with a hammer while turning the screw. They use them for motorcycle cases with Phillips screws that otherwise would never come loose with hand pressure alone. It you could rent one for a few dollars it would save you some money.

Your next step is to drill out the screw and re-tap the faucet stem.

Good luck!