This might look silly but what about running this container in lxc mode and just let it to have access to some specific devices so that it can chrome can run with sandbox but restricted from touching anything else!
Like this example that I am using to have cuda inside docker:
docker run -ti --lxc-conf='lxc.cgroup.devices.allow = c 195:* rwm' cuda
Its not privileged, but have access to devices that I want.