I would expect them all to do equally well, because of good management, and committed parents, rather than because they are one or the other.
There's a simple reason why faith schools, particularly over-subscribed faith schools, will do better: they're selective. Not on ability per se, but on stable families who can plan in advance. That makes schools' lives immeasurably easier.
Consider your local CofE primary. You're pretty much assured a place if you plan a couple of years in advance and turn up dutifully at the associated church. That means you are people who plan education two years in advance, and are living in the area semi-permanently. Oh look: the middle classes. You're only slightly less assured a place if you plan a couple of years in advance and turn up dutifully at some other church (older, more complicated admission requirements had a "that does not have its own associated school" clause, but that's much less common now). That means who are people who plan education two years in advance and are living in an area semi-permanently. Oh look: the middle classes.
That's why CofE schools have, compared to their neighbouring community schools, lower levels of FSM, EAL, SEN: they select for people who can and will plan in advance and navigate the system. The middle classes.
If a school set its admission requirement as "you must turn up to four meetings out of six over the preceding two years and write a 100 words on the subject of your choice" they would get the same effect.