Add a keyboard using XKB that matches the Mac/OS X phonetic Russian keyboard.
See here for a Chrome extension based on this package to add this keyboard to Chrome OS.
Note: the XKB files are those from Ubuntu 16.04.1 LTS. Your mileage may vary
(i.e. the setup.sh script will tell you it can't install the new versions if
your original versions don't match these reference copies).