Running `export $PATH` will try to export a variable with a name equal to the **value** of `$PATH` (after word splitting and filename generation). That is, it's equivalent to writing something like `export /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin`. And since `/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin` is not a valid variable name, it fails. **What you want to do is`export PATH`.**
`export` (equivalent to `declare -x` when not called within a function) in Bash maps the shell variable to an environment variable, so it is passed to commands executed from now one (in child processes or otherwise).
To print the value of a variable safely and readably, use `printf '%q\
' "$PATH"` or `typeset -p PATH` to print its definition.