After looking through it again, I'm not sure if the SDK contains the full client gold source engine, but it definitely does contain the source files required for the Half Life "addon" to the engine.
In the full version of the SDK (which has extra documentation over the one posted above), there is a file called ReadMe.txt, In that it says:
Single-Player Source:
=====================
The directory containing all the code required to rebuild hl.dll.
All games running on that engine work as .dll files and the engine itself is unmodified between them. Half Life is no exception. But, without the gold source engine itself, then being able to compile hl.dll (or .so on linux) is kind of useless. The SDK does contain the full server source, which should have most of the gold source engine, since the world simulation is all done server side. The server source does not contain any rendering code, for obvious reasons.
The SDK does contain the source required for the Half Life specific .dll to be compiled. So, if the full SDK contains the source to the client side gold source engine, then this is doable, which vgui.dll being the main problem. Otherwise, the engine itself needs to be mostly rewritten (I say mostly because the source to the server is there and it SHOULD contain the majority of the gold source engine).
EDIT: And attempting to use the Quake engine code might help with rewriting the engine if it comes to that, but the majority of the code to the gold source engine is not the same as the Quake engine.