I think the spirit magic system could easily be translated pretty directly by keeping the mechanics and just change to way things are worded.
Instead of calling an ancestor of great power, you call on the power of the angel Michael, or the demon Azazel, who were renowned in the heavens for their great skills and give you their power. Instead of having your fetch, you have your personal guardian angel, or bound demon. Instead of controlling bears by the power of nature, you have a demon of the beasts control them for you and then control the demon. Instead of controlling fire elementals, a demon just manifests as a powerful fire.
When you are spirit walking, perhaps you aren't walking around the "real" world looking for spirits, but more like your mind is walking around heaven/hell looking for subjects, all the while your physical form does nothing but speak in gibberish and draw sigils everywhere.
If you want to make it more risky, perhaps demons could be stronger, but involve a problem like incurring some sort of insanity for a few days if you fail a test with them.