Sorry you're having such a rough time OP! It's horrid how some families can be when there is very real love and support that's needed.
Bottom line: ultimately you live with your child and have her best interests at heart.... While your family presumably feel they, too, have her best interests at heart... But you know her best!!
To me, your brother is doing the classic 'minimising' of someone else's, to you quite obvious, problems.
Presumably, he is seeing special needs education as an absolute stigma, with no possible advantages to your daughter??
I have no specialist knowledge of speech difficulties, but I do know clinically, across the board, similar clinical presentation (ie what is seen), can have different underlying causes, that often will require different specialist input. Or sometimes just watchful waiting, essentially waiting ans seeing what happens but at the same time closely monitoring them.
Your brother, it seems to me, is assuming that your DD's difficulty is something that will gradually 'reverse' ie as a maturational, developmental change. It may be, but conversely, it may be something that needs extensive professional input.
However, unless he is some sort of education specialist professional who has conducted extensive clinical interviews and testing with your daughter (for example ed psy/speech and language specialist,)... I assume not
.... he will have diddly squat understanding and knowledge of your DD's difficulties, and indeed the depth and breadth of these difficulties.
It is a classic mistake to make, assuming that something that 'looks' quite simple has a simple and easy-to-remedy cause.
There is a reason these professionals take many years to train 
. If it is what that easy these decisions would be left to Joe Bloggs walking down the high st. He is just grandstanding.. (and bullshitting), which is crap and upsetting for you.. And ultimately, very unhelpful.
Years ago I worked in a special needs unit, it was interesting- the parents who were most against their child 'being somewhere like this' and the most vocal (actually a very good and well- respected unit with extensive trained special needs teachers wirh visiting specialist professionals ).... were often men, including grandparents /uncles etc... And to a man, they often had very little contact with the kid, eg a parent who had disappeared for years, or someone who saw their kid one Sunday a month to go to the park. They were largely clueless to the childs difficulties.( I hope this isn't too sexist, but it is my observation). I know that some of these men went to great lengths to hide their child's attendance at a special needs place. Some of the more hands on male carers would often sabotage appointments eg with ed psys, speech and language professionals as they wanted what THEY wanted, presumably a 'normal' child, without any recourse to the child's needs.
As you'll know, special needs education and inclusive education and all the issues around this, has been a hot potato for many years. There have been many advantages to inclusive education, but equally disadvantages.
Ultimately, you will have a strong idea of how your daughter will cope in each setting... Yes special schools are a different beast, but afaik they still have to abide by the national curriculum.
It is complete and utter bollocks for anyone to say the reason the LA are not funding is due to your daughter not needing this provision.... If this was the case, they would make reference to an educational psychologist and also any other specialist input professional, expert, reports.
Do what you think is best for your kid!. Hope you get to enjoy some sunshine on the BH!
Sending you a hug!