"There are three of us, including me."
Translation:Nous sommes trois, dont moi.
The problem is that this is a literal translation from English and the French wouldn't express it that way. "There are three of us" would more simply be translated to "nous sommes trois". Any emphasis on including yourself into it (moi y compris) would make no sense here, "nous" underlines this well enough to begin with. If you really wanted to say something that comes close to the meaning and emphasis of your original phrase, you'd rather say something like: nous sommes trois ici, et j'en fais partie (we are three here, and I am part of it), or: nous sommes trois ici, et je suis l'un d'eux (we are three here, and I am one of them).
un is also a noun, just as deux, trois etc are also nouns. As you are aware, every French noun requires an article even if we don't always translate it in English. This is particularly the case here, so un is not mistaken for the indefinite article.
Hence: Je suis l'un d'eux → I am (the) one of them.
"Including" is not a verb in this sentence - it is a preposition.
JH (the first comment) has confirmed that "moi y compris" was accepted. After going through the story of "Décisions" I came across a phrase "Y compris de notre mariage", I have a hunch that it is correct to say "nous sommes trois, y compris de moi." Could someone comment on this?