Championship is run by the Football League.
Premier League is run by the FA.
No it isn't.
Well if it isn't then perhaps you could enlighten me as to who runs what.
The Premier League runs the Premier League. The Football League runs the Football League. The FA sanctions all competitive games at national level, regardless of the league.
I've just checked the FA rules on advertising on shirts and it does allow sponsorship on the back of shirts so it must be a Premier League rules that prevents us from doing it.