I suspect that the cheapest way is to pay to enter the exam yourself, as long as you can find a school to host. Your daughter would then have to revise and study on her own, but it seems to me that this would be the minimal cost if it worked.
Following from clover's post, it might also depend on whereabouts you/your daughter lives. A quick search suggested that colleges offering general GCSE resits are not all that common, or certainly not free, but of course searching around for more than a few minutes might turn up something more attractive.
In the end, though, I don't think that some amount of cost can be avoided once you've left the state education system.
I am an ex-teacher. I could tell you some horror stories about other teachers I have had the misfortune to come across. Way back in 1991 I accompanied a party of children to France. On our last night we were rudely disturbed by a party from London which had just arrived. We were having a quiet drink in the bar when this very voluble cockney lady forced herself into our company. She freely admitted that she could not speak a word of French, she had just come along for the ride. She asked me where we had been that day so I told her that we had been to Rouen.
'Why there'? she enquired.
I explained that it was an ancient city, noted for the martyrdom of Joan of Arc.
I was lucky enough to have some excellent teachers when I was at school. One piece of advice we were given by a lady teacher was 'Treat everyone the way you would like them to treat you.' I've always tried to live up to that.
I won't bore you with the many run-ins I've had with 'English' teachers. When one told me that I should not say, "It's me, but it is I", I asked if she had ever heard of the disjunctive or emphatic pronoun. She hadn't a clue what I was talking about.