I know you already reached a conclusion on this, but here's a thought to chew on for the future.
Regarding the technicality of it, I think styles typically can't be copyrighted. The trademark "S" and slogan, as you mentioned, obviously will be copyrighted. However, I don't believe most pieces can claim a style or concept of design.
However, my end solution would be this: use a different design, and here's why. I have significant experience working with churches in regards to graphic design (in fact, I am currently employed at a church doing graphic design. I have come to be frustrated with Christian design always piggybacking off of famous designs. To me this is "regurgitating" someone else's idea. It's not really thoughtful design work, but kind of playing off someone else's design with a few tweaks. It really robs the respect or admiration for the design, because the designer didn't do the concept themselves.
I'm not critiquing you as a designer. Obviously I don't entirely understand your situation, but from what you've posted, it seems like the pastor or whoever is giving you direction is playing the role of the designer. They are telling you what to do because you have the skill set to make it a reality. I've worked with people like that before, and it's rough. But at some point it might be appropriate to step in and say that as the designer, you would like to pitch a few design ideas instead of merely doing what they want. That will likely result in a better product and outcome, even though this person might be really convinced that their idea is exactly what they want. Sometimes non-designers don't know what they want until they see it. If you pitch them something sharp, snappy, and original, chance's are they'll go for it. Even non-designers can tell when a design is gold. Besides, it just might save you from being stuck with the cross and starburst :)
Best of luck!