BTTV does work with third party routers, it certainly works with TP Link devices, you need to enable IGMP Snooping.
I doubt that the AC1200 wifi extender would be able to carry the multicast data.
Try connecting the YouView box directly to the router.
From the manual, it would appear you need to uncheck the IGMP Proxying (rather than disabling) as well as selecting BT Proxying. (Never sure of the difference between snooping and proxying)
Enable British Telcom IGMP Proxy Support
You must use this feature only for British Telcom (BT) Internet Protocol television (IPTV) service. It might
also work with TalkTalk IPTV service.
To enable BT IGMP proxy support:
1. Launch a web browser from a computer or WiFi device that is connected to the network.
2. Type http://www.routerlogin.net.
A login window opens.
3. Enter the router user name and password.
The user name is admin. The default password is password. The user name and password are
The BASIC Home page displays.
4. Select ADVANCED > Setup > WAN Setup.
The WAN Setup page displays.
5. Clear the Disable ITMP Proxying check box.
6. Select the Support British Telecom (BT) IGMP Proxying check box.
7. Click the Apply button.
Your settings are saved.