Just because you value swimming and cinema trips, it does not mean that other families do.
There is a strong generalisation in your post regarding the Muslim families you know at your child's school and their immigration status.
I know many, many Muslim families who are second and third generation Asian British. They did not choose to live in the UK, they were born here. It was their parents and grandparents who made such decisions. Their children are educated in the state system and have further evening or Saturday classes studying Arabic and the Koran.
Over the years I have known families of Portuguese, Spanish, Japanese, Chinese and Ghanian descent who send their children to additional classes related to their family language and heritage. This has nothing to do with wanting to not intergrate and everything to do with wanting to maintain your family's cultural links.
I have often said that if I was to move to another country for work whether it be France, China, Bagladesh or America, I would still be English in my heart. Even if I took citizenship elsewhere, my roots will always be English. So I would want my children to learn about their heritage, culture and language. So most likely, they would be off to English school on a Saturday.
Part of being an inclusive community is welcoming others but accepting difference. Just because you want play dates and swimming, it does not mean that it is high on the agenda of other parents.
I'm sure your dd can learn to accept that she will do different things with different friends.