Here's the general solution for when you want a verbatim value in a command line but don't know how to quote it:
Put your value in a file exactly the way you want it:
$ cat file
PROMPT_COMMAND='echo -ne \"\033]0;${USER}@${HOSTNAME}: ${PWD}\007\"'
Use printf to automatically escape it:
$ printf "echo %q\n" "$(< file)"
echo PROMPT_COMMAND=\'echo\ -ne\ \\\"\\033\]0\;\$\{USER\}@\$\{HOSTNAME\}:\ \$\{PWD\}\\007\\\"\'
The resulting command produces the same result as the file, with all special characters and non-trailing line feeds intact:
$ echo PROMPT_COMMAND=\'echo\ -ne\ \\\"\\033\]0\;\$\{USER\}@\$\{HOSTNAME\}:\ \$\{PWD\}\\007\\\"\'
PROMPT_COMMAND='echo -ne \"\033]0;${USER}@${HOSTNAME}: ${PWD}\007\"'
Important things to note:
You would do this once, interactively, to generate the echo command to use. The suggested answer is NOT to create the temporary file on each of the 4300 machines.
It's bash specific, and so is the command it produces.
If you use ssh, you need a separate layer of escaping.