I have an Apple adapter to connect my iPad to a TV via the Component connections. Apps need to have video out enabled to view them on the TV. Many of the Apple apps, plus BBC iPlayer and the ITV player have this enabled but the BT Sports app does not. It cannot be to protect rights because, if I connect my iPad via an adapter to the D-sub 15pin connector on the TV (as you would connect a computer to it), full mirroring of anything on the iPad screen is shown on the TV. The Component connector is, however, more convenient and gives sound as well - the PC connector route requires an additional audio lead. Why doesn't the BT Sports app follow the example of BBC iPlayer and enable video out?
The TV has HDMI sockets but, when I bought my adapter from Apple, I don't think that there was an option of a cable with an HDMI connector and, in any case, my iPad doesn't have a lightning connector. The issue is that the BT Sports app has limitations on an iPad that other similar apps do not have.
Sorry I can't answer your question with regard to the cable connection. However the best, easiest but unfortunately a bit more expense, is to get the AppleTV box. The App works perfectly with this on all three channels. A new one is £99 but they occasionally appear in Apples online refurb store for £85. They will be as good as new and you still get the full years Apple warranty with the refurb products.
It also means that you can take your iPad and the ATV to any location with a wifi connection and TV and use it there too.