I want to make the Search in Google context menu to open my default browser instead of Safari.
I believe "Search in Google" is a Mac OS X service provided by Apple. Maybe since they don't know how to pass a webpage in parameter for every browser in the wild, I suppose they limited it to Safari.
What you might try, is to develop your own Service, then go to
System Parameters > Keyboard > Services
Disable the "Search in Google" by Apple and replace it with your own.
You could also try to edit Apple's Service and replace Safari with the browser you want. Since I'm not on my Mac right now, I can't search for them or look if you can edit them.
This worked for me. It shows how to create your own service that is independent of Safari. https://superuser.com/questions/369934/mac-os-x-lion-chrome-shortcut-for-search-with-google