I’m only saying this for your awareness, not to attack, but you are very naive. Culture isn’t just things like drinking, praying, food eaten - it’s also family dynamics, obligation. In favour based cultures, individual autonomy is overridden for the greater good of the family unit or social value. It’s complex to navigate as a westernised Muslim who grew up in the culture, let alone someone who has no experience of it. You’re already experiencing the impact of obligation and not discussing or engaging with you, they’re not even doing it with malicious intent which is why you can say they’re really nice, but it’s just default to them not to consult or engage, and it is partly misogynistic and disrespectful to you as they don’t see your input of importance.
So I’m saying you’re so you can own it and recognise this situation for what it is and start setting up really strong boundaries.
Even without the cultural elements, you need to set appropriate boundaries. You’re not a blended family, he is your ex and their should be contact agreements in place.
Don’t worry about your daughters missing out, you and they will make many friends of different cultures that you can mix with on your terms.
It’s really important your daughters have a strong and stable identity and home with you, especially as a result of their mixed heritage.
If you feel you don’t know how to approach all this save face business and find it awkward, it will be even harder for them, as they may be made to feel rejected if they don’t present or act “Muslim” enough. It’s not just about “seeing a different culture”. You need to be aware of the dynamics so you and your daughters don’t suffer from them. Obligation and shame, such as you’ve already experienced with the lying, not telling you, saving face coming above everything else can be used to have a huge impact on you. You don’t want saving face to come above your daughters.