Quite a while ago , OR asked the regulator Ofcom, if they could interact directly with end users in situations like yours , but Ofcom , after asking the likes of Sky and TT their opinion, said No, those companies only want you to contact them, and only your provider to contact you, so, as you already knew, all inquiries need to be made to whatever provider you use.
Your fault could be a problem at the ‘exchange’, perhaps it’s not , but if it’s TT or Sky ( or company’s that use their equipment) then whatever your problem is , it cannot be ‘BT’, TT and Sky use their own equipment in the exchange, and Openreach to connect it to your home, so its nothing to do with BT’ in any meaningful way.
If your provider refers to Openreach as BT Openreach, then you could tell them to stop doing it as it’s unintentionally misleading at best, and designed to deflect blame at worst
