I quicker solution is to use another one of their free domains that’s already in the Public Suffix List (PSL).
But to answer your question:
Can it get an exception?
The short answer is NO, you can’t request an exception - only the owner of the domain (NOIP) can request an exception and only for special situations.
Even the PSL requests “Requests for changes to the PRIVATE section MUST come from the domain owner” See: https://publicsuffix.org/submit/ (Again NOIP)