I'm not sure what you mean by 'push' for the 11+'? If you mean, 'insist that she take the exams', then that is an issue for you to sort out with your DD. Does she actually want to take them, would she actually like to go to CCHS or any of the CSSE girls' grammars, etc? Neither the CSSE nor CCHS will turn down your request for her to sit their exam. However, if by 'push', you mean 'have intensively coached for', then please remember that just as a dog is not just for Christmas, gaining a grammar school place should imply that you are able to keep up with the pace of lessons once you are there. Looking at your DD's levels, although the English may be a bit of a struggle, if she is still progressing rather than having reached a plateau, she may be okay. Have you got her to have a look at the practice papers available from the CSSE?
Also, what are your intentions for her if she doesn't take / doesn't do well enough in the state 11+ exams? As you say that your school both teacher and usually grades VR, I assume that your DD is currently at an independent school? Does it have a senior department that the school is trying to get her to stay on for, rather than lose her to the state sector?
Outside of a dog, a book is a man's best friend. Inside of a dog it's too dark to read.Groucho Marx