"I’ll probably write an elisp function to send the output to GPT and insert the results into the buffer"

I suggest taking a look at my LLM command-line tool. It's great for cobbling together these kinds of things, because you can use whatever shell integration your environment has to pipe things to it.

    cat bad-dictation.txt | \
    llm -m gpt-4-turbo --system '
    You are going to correct for text that
    has been produced by voice recognition
    software. Rewrite any text provided to
    you. The text will not have punctuation,
    so please add where needed. If the text
    has the word period, then insert a
    period, and do not insert another one
    just before or after. If the text has
    “new line” or “newline”, insert a newline
    character and start a new paragraph. If
    the text says “comma”, insert a comma. If
    the text has a number spelled out,
    replace it with the actual number. So
    “five thousand” becomes “5000”. Also, if
    something seems off in the text, it was
    probably due to a misrecognized word.
    Please correct for it.' > out.txt

Should be easy to call that from elisp, and you can then install plugins to have it talk to other models like Claude or Llama 3: https://llm.datasette.io/en/stable/plugins/directory.html