Given any terminus in London looks like it would be expensive, what about the option of not having a terminus – building a tunnel and connecting the south western line to HS2, and then having mainline trains running across the country, as in France and Germany?
Through services from the south coast and Brighton to the North probably wouldn't be worth doing it for its own sake, but if a tunnel turned out cheaper than having to create a major terminus in central London it might be an option?
You would still have to have a station in central London but it could presumably be smaller if there was no need to terminate, and fit in more places.