Take a look at how pacapt detects the OS:
# Detect package type from /etc/issue
_found_arch() {
  local _ostype="$1"
  shift
  grep -qis "$*" /etc/issue && _OSTYPE="$_ostype"
}
# Detect package type
_OSTYPE_detect() {
  _found_arch PACMAN "Arch Linux" && return
  _found_arch DPKG   "Debian GNU/Linux" && return
  _found_arch DPKG   "Ubuntu" && return
  _found_arch YUM    "CentOS" && return
  _found_arch YUM    "Red Hat" && return
  _found_arch YUM    "Fedora" && return
  _found_arch ZYPPER "SUSE" && return
  [[ -z "$_OSTYPE" ]] || return
  # See also https://github.com/icy/pacapt/pull/22
  # Please not that $OSTYPE (which is `linux-gnu` on Linux system)
  # is not our $_OSTYPE. The choice is not very good because
  # a typo can just break the logic of the program.
  if [[ "$OSTYPE" != "darwin"* ]]; then
    _error "Can't detect OS type from /etc/issue. Running fallback method."
  fi
  [[ -x "/usr/bin/pacman" ]]           && _OSTYPE="PACMAN" && return
  [[ -x "/usr/bin/apt-get" ]]          && _OSTYPE="DPKG" && return
  [[ -x "/usr/bin/yum" ]]              && _OSTYPE="YUM" && return
  [[ -x "/opt/local/bin/port" ]]       && _OSTYPE="MACPORTS" && return
  command -v brew >/dev/null           && _OSTYPE="HOMEBREW" && return
  [[ -x "/usr/bin/emerge" ]]           && _OSTYPE="PORTAGE" && return
  [[ -x "/usr/bin/zypper" ]]           && _OSTYPE="ZYPPER" && return
  if [[ -z "$_OSTYPE" ]]; then
    _error "No supported package manager installed on system"
    _error "(supported: apt, homebrew, pacman, portage, yum)"
    exit 1
  fi
}
As you can see it first checks /etc/issue, then failing that the script looks for the associated executable file for each package manager.
But heck, why not just use pacapt, instead of rolling your own?