You actually head-canon Santa.
Yes.
Are you considering to add him in the SCP mythos?
Eventually, but probably not as an SCP.
Saint Nicholas
Low-level Wondertainment type. Santa isn't him, but as a Wondertainment type, he can tap into the underlying Wonder in the universe and do a kind of spirit-advisor channeling thing with Old Nick.
Krampus
Santa's partner/rival. In the places that Santa has less power, he goes in and tries to at least alleviate the suffering he finds there. Not so much like a relief worker, think more of a psychopomp. If you ignore the Dr. Wondertainment stuff, I like the portrayal in The Twelve Days of Site 87's Christmas.
Odin
Whole other can of mythological worms.
Basically, the various Nobody iterations of Santa have been around for who knows how long, and this one just happens to be a Wonder-maker. The way that Nobodies work, that means that he has always been a Wonder-maker. However the base concept that has stayed the same through all the iterations is that of protector (think North from Rise of the Guardians). The winter solstice is the darkest day of the year when dark things stir, but Christmas and other solstice holidays pertain to light beginning to overcome dark. So while out doing their rounds, Santa and Krampus do a bit of cleaning in the "things that go bump in the night" department. Oh, and the reason we never see Santa when he delivers presents is that the universe retcons so that they were gotten for us by other humans.