stuck screw now stripped when trying to remove outdoor faucet handle
You could drill a small hole straight into the screw from above and use a screw extractor (or &quot;ez-out&quot;), it can be a difficult operation with varying degrees of success depending on experience and skill.

Or you could break off the handle, you could then get a grip on the protruding screw with pliers to remove it. The problem is you may or may not be able to procure a replacement handle.
You should probably start looking at replacing the entire faucet just in case your efforts to remove the screw are unsuccessful.