This matches how the rest of the build is organized but also hopefully
solves an issue where the VS Code directory is empty because we try to
cache it directly and Travis might be creating it.
Also too the opportunity to rewrite the build script since there was a
change in the build steps (mainly how the product JSON is inserted) and
to get the build changes out of the patch. It also no longer relies on
external caching (we'll want to do this within CI instead).
