It's a tricky one. The source of the legislation is the Local Government Finance Act 1992. I had a quick look at this but it is written in the way that it is very unspecific about how local authorities should apply discounts.
I suspect that the best opportunity to avoid the charge is via the London Council, not the W Midlands, on the grounds that the second occupant maintains a permanent home elsewhere on which CT is paid, and he is only staying at the London property with his friend on a temporary and non-full-time basis.
If you tell us which London Council is involved someone will be able to check what they specifically about discounts - council policies do vary.