How does "_ and I"/"_ and me" rule change when inside brackets?

This question was specifically motivated by the Youtube video https://www.youtube.com/watch?v=iitXhgif_lo which has the title "How one little boat (and me) held up miles of London traffic".

At first, knowing that, without the brackets (i.e: "How one little boat and me held up miles of London traffic") the use of "and me" would be a mistake and you should use "and I" because the pronoun is still part of the subject of the sentence, I thought the use in this title was also a mistake.

However, I then checked the sentence against Microsoft Word's grammar checker and Grammarly, and I found that, while without the brackets, the error was detected, with the brackets, no errors were found (obviously grammar-checking software is never foolproof, but I find it usually works quite well in simple cases such as this).

After thinking about it more, the part is brackets is not really a part of the main sentence, and so I guess the "me"/"I" is not actually doing the verb, but it still sounds odd to me.

Both Grammarly and Word also seem to accept "(and I)" as well, so I wondered, is one of them more correct? Is there a difference in meaning between the two? Are Word and Grammarly completely wrong?

The war and the battle for It is I as opposed to It is me, has been going on for a long time. In 1877, Ebenezer Cobham Brewer wrote in the entry on “me” in “Errors of Speech and of Spelling, Volume 1”

‘Me” is used after the verb To be, and after the words than, but, like, and as, with such pertinacity it is at least doubtful whether it is not correct. C’est moi is the French Idiom, not Cest je, and It is me is far more common than It is I. (“Me” is dative not accusative case.)

Chaucer (Early Middle English) always used the usual nominative after "to be" and it is probably thus to the French influence and moi that we owe the nominative "me."

In "How one little boat (and me) held up miles of London traffic" "and me" is not part of the subject - it is in parenthesis.

Me as the nominative is commonly used in personal stressed pronouns (also known as disjunctive pronouns and emphatic pronouns.)

"Who wants £10?" "Me!"

Without the brackets: "How one little boat and me held up miles of London traffic" "and me" is no longer in parenthesis - it forms an equal part of the subject.

It is perhaps clearer in the present tense:

"How one little boat (and me) holds up miles of London traffic."

"How one little boat and me hold up miles of London traffic."

EDIT TO ADD

"How one little boat and I hold up miles of London traffic." sounds a little formal but is correct = "How we hold up miles of London traffic."