Since you probably won't be able to install facter on a remote server, you can mimic what it does to find the OS name. The Ruby code for the operatingsystem fact can be found here on pastebin. Basically, it looks through the different *-release files and others to determine the OS name.
Some of the files it looks at:
/etc/debian_version
/etc/gentoo-release
/etc/fedora-release
/etc/mandriva-release
/etc/mandrake-release
/etc/meego-release
/etc/arch-release
/etc/oracle-release
/etc/enterprise-release
/etc/ovs-release
/etc/vmware-release
/etc/redhat-release
/etc/SuSE-release
/etc/bluewhite64-version
/etc/slamd64-version
/etc/slackware-version
/etc/alpine-release
/etc/system-release
/etc/centos-release
I'm sorry if you find duplicates in this list, I produced it quickly with grep. Should be fairly easy (albeit a bit tedious) to port this to a POSIX shell script.