Douxie is a wizard. There’s no doubt.
First of all, look at his trajectory. He was either flying or has some other sort of magical/supernatural/alien talent. There’s no way, even in an animated show, that normal teenagers just fly through the air and can take down a GummGumm.
That’s not back lighting, that’s a flash of light. Light=magic, typically, so we can assume that Douxie is hitting the GummGumm with a bit of magic. Also, I think the guitar is also his magical staff. That’d be cool.
Linguistics! He talks as if he’s been raised in medieval Europe and, if you ask me, he seems rather chivalrous. “Lady Claire” kinda clenches it.
His presence at the restaurant was no accident. He knew Morgana was messing with Claire, which was probably why he was only interested in her (of that whole group of girls) in the first place. I doubt he actually works at the restaurant.