It sounds like the terms of your contract have been pretty much wrapped up, but if there's still a chance for further discussion, you could consider raising the issue of an education allowance so that you can send your son to an international school.
However, there are also lots of good reasons to go with a local school, especially since your son is still so young. If you're planning on staying in Beijing for the long term, Chinese schools are an excellent way to help expat children integrate and learn the language. The younger they are when this process takes place, the easier they tend to adapt.
Homeschooling is an option and is legal but might become complicated later (as in, much later) when it comes to university as the qualification needed to attend a Chinese university can't be obtained through homeschooling.
If you want a bit more info about schooling in Beijing, click here.
For fee costs of public schools I'd recommend getting in touch with them directly.
Hope this helps.