I know the school you mean (I think). If I'm right, it really was a very small school. Almost any other school is going to be a bit of a shock.
Would your daughter feel more comfortable entering a new school if she had some friends going with her? Personally, I'd stick with the independent school, if only because your daughter is likely to know a few of the other students, and they can all start over together.
Is the alternative independent school non-selective and only up to GCSE? That might account for the lack of science options.
A school with a sixth form would be a better bet, if you could find one.
Is there any chance of testing for/asking about a grammar school place?
The top sets of quite a few of the local schools are really very good, as they tend to be made up of children who didn't pass the 11+, but blossomed later.
The only people I know who switched to non fee-paying schools before 6th form, did it for financial reasons, and would have preferred to stay in the independent sector.
If your daughter is happy to give the comp a go though, then why not? I'm almost certain the fee-paying school would take her in later if it didn't work out after a term.