As a vocal supporter of the BBC (yes even with all it's many, many flaws) and am in favour in principal of the license fee, I don't think this is an appropriate way to fill the black hole in their finances.
It should in fact just become part of general taxation as that's the only way to make it fair. Even if you don't consume their content, it provides a national service, it projects soft power abroad, it's a primary news source for more than half a billion people and if that's worth anything (in my view it is) then it should come out of the central budget.