I use firefox, and find this function useful:

     # Open a new firefox instance, with a temporary profile
     function firefox_temp() {
        dir=$(mktemp -d)
        echo "Profile directory: ${dir}"
        firefox --new-instance --profile ${dir}
        echo "Cleaning up .."
        rm -rf ${dir}
     }

I'm not familiar with Chrome, but perhaps there is a similar "profile-directory" argument you can point to a temporary location?