SUMMARY
The best way to sync files (code, in my case) is to use git and bitbucket/github. Install git in the chroot, and sync the code into the Downloads folder so it can be accessed from both ChromeOS and the chroot (and compiled and ran locally on both computers). I stored all other files in Google Drive so they could be accessed from ChromeOS, other computers, and the chroot.
Here are the options I looked into, but did not end up using
Accessing the Google Drive cache directly on the chromebook
Google drive in the chrome os stores the file data in
/home/chronos/user/GCache/v1/files
However, it does not name the files how they are named in google drive (it names them by some uuid, and stores meta in another folder).
3rd party tools to mount Google Drive folder in linux
https://github.com/dsoprea/GDriveFS
I was able to get GDriveFS working, but it was slow and chmod does not work in its file system. The permission are all static at 666, so programs will not execute.
https://github.com/astrada/google-drive-ocamlfuse/
I was not able to get ocamlfuse working on an arm Chromebook.
https://github.com/Grive/grive
I was able to get Grive working, but it has some problems and hasn't been updated in over a year. Would not recommend if there is a chance of merge conflicts.