I understand that an ethernet cable connection is not your preferred connection method, but it must be worth trying, just temporarily, to see if the pixellation ceases. If it does, you'll know that there is something that needs fixing with your WiFi - and this is not BT's/EE's responsibility. If it doesn't, then you'll be armed with the knowledge that there must be something wrong with the TV Box or with the broadband connection.
Just get a very long ethernet cable and string it up between router and TV box.
If the ethernet cable connection does eliminate the pixellation then you could also consider powerline connectors as an alternative to WiFi.
