You guys understand that I'm not asking for software to be built from scratch, right? I only need some modifications to existing software such as SMF or phpBB. A few of the required modifications are quite extensive, but many are just little tweaks. The specification document is long not because the job is huge, but because I included a lot of detail.
What do you mean "find some well written PHP forum software"? I can't think of any forum software, free or commercial, that wouldn't have a license agreement forbidding you to modify and sell/resell it.
Transisto's link contains a long list of such software.