Gosh that's a hard one!
Is it simply not going to the school her mates are going to? If so it's a non issue. If they are decent mates then they will keep in touch - fbook etc if you'll let her (at just 11 I wouldn't) but txting etc and can meet up loads- they'll still be the kids living near to her.
Our situation was different I guess. Big city , even the local community schools are some mixed,some single sex as well as small catchments, so it was normal for kids to go to different schools even taking selection out of the mix.
As it stands that is the school place she has, and for what it's worth the fact she got a place means it's academically the right place for her. It is the school she will start in September. She isn't going to get a waiting list place now and it would be unfair to the kids who really need the comp place to fight them now- simply rejectin the grammar place would be foolish as she'd get allocated "sinksville" if the LA gave her a place at all.
IME (and remember I only have grammar school kids so a one sided view) the friendship/team building stuff they do means that a few weeks into term they have friends (and know lots of the other kids too) . These friendships have endured like no other (apart from DD1 and her best mate who were welded together at the hip at 7 and went through grammar together too, aquiring 3 other "best mates" as well). DS didn't really have friends till he went selective and DD2 is everyones mate anyway. Kids at our grammars often are the only 1 from their school attending let alone their class.
I think you need to have a chat with her and agree that maybe if she isn't happy come the summer, having given it a year- seen the ups and downs of friendships, grammar level working etc, you will put her down on the waiting list for the faith school. Only agree this if you will carry it through though.
IME she'll not look back after the first week or so in September BUT equally if she did stick a year and was still miserable I'd grit my teeth and do what she wanted after looking at the pros and cons. 5yrs is a hell of a long time for a teen to be miserable if you look at it that way.